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II. Absolute Measurements of a Resistance by a Method based on thato/Lonimz. 

By F. E. Smith, A.RC.Sc. 

(From the National Physical Laboratory.) 

Communicated by R. T. Glazebrook, M.A. y F.R.S. 

Eeceived October 27, — Read December 11, 1913, 



Principal Contents. 

rage 

Historical note on the absolute measurement of resistance . 28 

Introductory 33 

Method of Lorenz . . 35 

General description of instrument 38 

Motor and fly-wheel 40 

Rotating discs and their supports 40 

Rotating conducting wires .41 

Contact brushes and their lubrication 42 

Coil supports 46 

Lubrication of bearings . 46 

Regulation of speed , . 47 

Recording the speed 47 

Magnetic tests 49 

Construction and measurement of coils 51 

Diameters of coils 53 

Axial lengths and pitch 61 

Erecting and adjusting the instrument 64 

Insulation tests 65 

Effective diametral distance of segments 66 

Measurement of distance between the coils . 68 

Calculation of mutual inductance * 70 

Correction for conicality . . . . , 76 

Correction for variation in pitch 79 

Effect of iron of motor . . 81 

Arrangement of circuits . 83 

Standard resistances used 86 

Setting of coils to be co-axial 87 

Determination of resistance in absolute measure 93 

Probable errors 99 

Conclusions. 107 

VOL. OOXIY. A 510. E 2 Published separately, March 19, 1914. 



28 MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

Historical Introduction, 

There are many methods by which a resistance can be measured absolutely in the 
electromagnetic system of units, and all of these necessarily involve absolute measure- 
ments of a length and of a time. The length may be the axial length or radius of a 
coil, or the radius of a disc, or it may involve all of these, and the time may be the 
time of vibration of a magnet, or of a rotation of a coil, or of the period of an 
alternating current. In any case, the precision secured in the measurement of a 
resistance depends primarily on the accuracy obtained in these measurements of 
length and of time. 

The first absolute measurements # of a resistance were made by KiRCHHOFFf in 
1849, but it is to W. Weber J that we owe the first distinct proposal (in 1851) of a 
definite system of electrical measurements according to which resistance can be 
measured in terms of an absolute velocity. 

Weber devised three methods by which the resistance of a wire can be determined 
absolutely, and he published the results of experiments by two of these. The first 
was by means of an earth inductor, and the second by observing the damping of a 
swinging magnet, the results obtained differing among themselves by 5 parts in 1900. 
In 1853§ he made a determination of the specific resistance of copper, but the experi- 
ments were made more to develop the methods than for exact measurements. 

In 1862 Weber 1 1 made a more exact determination of resistance using a method 
compounded of his first two methods and eliminating the constant of the galvano- 
meter. The results of these experiments were embodied in a determination of the 
value of the Siemens unit and of a standard coil sent to him by Sir William 
Thomson, but the unit obtained was about 8 per cent, less than the 1863 unit of the 
British Association Committee on Electrical Standards. 1T 

The measurements made by Maxwell, Fleeminc Jenkin, and Balfour Stewart 
for the British Association Committee, although subsequently found to be incorrect 
by nearly 1*5 per cent., are the first with any claim to precision. The method 
adopted by these experimenters is that of the rotating coil and was devised by 
Prof. Thomson (later Lord Kelvin) independently, we believe, of a prior suggestion 
by W. Weber. The apparatus consists of a short-circuited coil rotating about a 

* 

* Admirable summaries and criticisms of various methods will be found in Wiedemann's ' Electricitat/ 
vol. IV., p. 910; Mascart and Joubert's * Lemons sur PElectrieite/ II., p. 581; Lord Rayleigh, 'Phil. 
Mag./ 1882; Wiedemann, 'Phil. Mag./ 1882; Glazebrook, 'Electrician/ 1890; Dorn, ' Wissenschaft- 
liche Abhand. der Phys. Techn. Reichsanstalt/ vol. II., p. 357, 1895. 

t "Bestimmung der Constanten von welcher die Intensitat inducirter elektrischer Strome abhangt," 
<Pogg. Ann./ Bd. 76, S. 412. 

J ' Elektrodynamisehe Maassbestimmungen/ or ' Pogg. Ann./ Bd. 82, S. 337. 

§ ' Abh. d. Kon. Ges. d. Wissenschaften zu Gottingen/ Bd. 5. 

|] 'Zur Gal vanometrie/ Gottingen, 1862. 

51 'Eeports of B.A. Electrical Standards Committee/ 1863. 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 29 

vertical axis in the earth's magnetic field. Currents are induced in the coil and these 
produce a deflection of a small magnet suspended at the centre. The dimensions of 
the coil and the time of its rotation being known, the resistance of the wire of 
the coil can be calculated. This method was also used by Lord Rayleigh and 
Prof. Schuster* in 1881, by Lord Rayleigh t in 1882, and by W. WeberJ also in 
1882. The results are given in Table I. 

In the method of the earth inductor due to W. Weber a coil is mounted on a 
vertical axis with the mean diametral plane in the magnetic meridian. In circuit 
with the coil is a ballistic galvanometer through which a quantity of electricity flows 
when the coil is quickly turned through half a revolution. The resistance of the 
whole circuit can be calculated from the dimensions of the coils, the constants of the 
galvanometer, and the deflection produced by the half rotation of the coil. Methods 
based on this inductor principle have been used by F. Kohlratjsch and G. Wiede- 
mann. § 

In a third method used by Kirchhoff, two coils, between which, there is a mutual 
inductance, are joined up in series with a battery and galvanometer, and a resistance, 
R, joins the junction of the two coils to a point on the circuit between the galvano- 
meter and battery. The steady current deflection of the galvanometer is first 
observed and then the throw due to one of the coils being removed to a position in 
which the mutual inductance is zero. The dimensions of the coils and the constants 
of the galvanometer enable the resistance R to be calculated. Methods based on this 
inductive principle have been used by Rowland, || Glazebrook,1T and Mascart, 
de Nerville, and Benoit. ## Many of the developments of the method are of 
extreme importance, and, except for the essential principle being the same, there is 
little in common. Similar remarks apply also to the developments of other methods. 

The method adopted by RoiTift and HimstedtJJ is somewhat similar to that of 
Kirchhoff, but, instead of having to observe the deflection due to a single impulse, a 
constant deflection due to a series of impulses is obtained. The current through one 
of the fixed coils is made and broken n times per second, and the galvanometer 
circuit is made only on the make or break of the current. The method adopted by 
Guillet §§ belongs in part to this class and in part to the method suggested by 

LlPPMANN. 

* Eayleigh and Schuster, 'Roy. Soc. Proc./ vol. 32, 1881. 

t Eayleigh, 'Phil. Trans.,' vol. 173, 1882. 

J Weber, ' Der Eotationsindukfcor,' 1882. 

§ G. Wiedemann, £ Abh. der BerL Ak.,' 1884. 

|| Eowland, 1876, 'Physical Papers,' pp. 145-239; The Johns Hopkins Press, 1902. 

H Glazebrook, 'Phil. Trans.,' vol. 174, p. 223, 1883. 

** Mascart, de Nerville, and Benoit, 'Ann. de Chemie et de phys.,' VI., p. 1, 1885. 
ft Eoiti, 'Nuovo Cimento/ III., 15, 1884. 

It Himstedt, 'Berichte der Naturforschenden Ges. zu Freiburg i. B.,' Heft I, 1886. 
§ GmLLftr, ' Journ. de Physique,' 8, pp. 471-477, 1899. 



30 ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

Another method used by Weber is commonly called the method of damping. A 
magnet is suspended within a coil and set in oscillation (a) when the circuit is open, 
and (b) when the circuit is closed. The periods and logarithmic decrements are 
observed, and from a comparison of the results the resistance of the coil can be 
calculated. This method was used by Weber and with important modifications by 
Dorn, # WiLD,t and Kohlrausch.J 

In the method due to Lorejstz§ a metallic disc is rotated at a constant rate in a 
magnetic field produced by a current which circulates through a coil co-axial with 
the disc. The disc is touched at its circumference and centre by two wires, and the 
difference of potential is balanced against that at the extremities of a resistance, R, 
the current through which is the same as that circulating through the coil. When the 
mutual inductance of the coil and disc circumference, and the rate of rotation of 
the disc are known, the resistance R can be calculated. Methods based on this 
principle have been used by Lorenz, || Lord Rayleigh and Mrs. Sidgwick,! 
Rowland and Kimball, ## Duncan, Wilkes, and Hutchinson, ft Jones,JJ and 
Ayrton and Jones.§§ 

Foster, 1 1 1| and afterwards Lippmann,!! suggested the use of a rotating coil, but 
contact with the extremities of the coil was made only at the moment when the 
induced voltage was a maximum. The induced voltage is balanced by that due to an 
external current (which may produce the field in. which the coil rotates) through a 
known resistance. Observations by a method based on this principle have been 
carried out by Lippmann and Wuilleumier. ### 

The most recent determination of a resistance in absolute measure is due to 
A. Campbell. ttt In Campbell's experiments two very nearly equal alternating 
currents in quadrature, taken from a two-phase alternator of sine-wave voltage, are 
passed through a resistance R and the primary circuit of a variable mutual 
inductance respectively. The ratio of these two currents is measured by passing 
them through equal resistances and comparing the deflections on an electrostatic 

* Dorn, ' Wied. Ann.,' 17, 1882, and 36, 1889. 

f Wild, 'Mem. de l'Ac. des Sc. St. Petersburg,' tome 32, Nro. % 1884. 

X Kohlrausch, ' Abh. der bayr. Ak. d. W./ Bd. 16, 1888. 

§ Lorenz, 'Pogg. Ann.,' 149, p. 251, 1873. 

|| Lorenz, 'Wied. Ann.,' 25, p. 1, 1885. 

H Lord Kayleigh and Mrs. Sidgwick, 'Phil. Trans.,' vol. 174, p. 295, 1883. 
** Eowland and Kimball, 'La Lumiere Electrique/ vol. 26, pp. 188, 189, 477, 1887. 
tt Duncan, Wilkes, and Hutchinson, 'Phil. Mag./ p. 98, 1889. 
%X Jones, 'Electrician,' p. 552, 1890. Also 'B.A. Electrical Standards Reports/ 1893, 1894. 

§§ Ayrton and Jones, ' B.A. Electrical Standards Reports/ 1897. 

Jill G. Carey Foster, 'B.A. Electrical Standards Reports/ 1870. 
UH Lippmann, 'Comptes Rendus/ 95, p. 1348, 1882. 
*** Wuilleumier, 'Journal de Physique/ 11, 9, p. 220, 1890. 
ttt Campbell, 'Roy. Soc. Proc./ 87, 1912. 



EESISTANOE BY A METHOD BASED ON THAT OF LORENZ, 31 

voltmeter placed across either resistance. The voltage across the resistance R is 
balanced (by the help of a tuned vibration galvanometer) against the voltage induced 
in the secondary circuit of the mutual inductance. The value of the variable mutual 
inductance is found by comparison with a fixed mutual inductance whose value is 
calculated from its dimensions, and the resistance R is determined in terms of this 
inductance and the frequency of the alternating current. 

Table I. gives the principal results, the values given in columns 5, 6, and 7 being 
those given by the experimenters. It will be seen that in some cases mercury 
standards of resistance were available, and in other cases the results are given in 
terms of the British Association unit (B.A. unit) or the Siemens unit. When 
mercury standards of resistance were available, the results (previous to 1892) state 
the length at 0° 0. of a column of mercury having a uniform cross-section of 
1 sq. mm. and a resistance of 1 ohm. The Siemens unit of resistance is the resistance 
at 0° C of a column of mercury 100 cm. in length and 1 sq. mm. in cross-section; 
results which give the absolute value of the Siemens unit may therefore be reduced 
to give the length representing 1 ohm by taking the reciprocal of the absolute value 
and multiplying by 100. 

In 1892, and again in 1908, the international ohm was defined as the resistance of 
a specified column of mercury. In 1892 Dr. von Helmholtz pointed out that a 
difficulty arose in determining the cross-section of a column of mercury owing to 
there being some doubt as to the correct value for its density. He suggested that 
the difficulty should be avoided by stating the mass of a mercury column of a given 
length which has a resistance of 1 ohm. This was agreed to and the international 
ohm was defined as the resistance at 0° C. of a column of mercury 14*4521 gr. in 
mass and having a length of 106*3 cm. The number 14*4521 is the product of 1*063 
and 13*5956, the latter number representing at that time the mean of the best 
determinations of the density of mercury at 0° C. The cross-section of the specified 
column is therefore equal to 1 sq. mm. or nearly so. The ratio of the international 
ohm to the Siemens unit may therefore be taken as 1*063. 

The B.A, unit is so much referred to in the earlier determinations that it may be 
useful to state clearly what is meant by the unit. In 1864 Messrs. Matthiessen and 
Hockin constructed a number of coils of various materials to represent at certain 
specified temperatures resistances of 10 9 cm./sec. units of resistance as determined by 
the 1862-3 British Association Committee on Electrical Standards. The resistances 
of these coils did not keep absolutely constant, and in after years the B.A. unit was 
taken as the mean of the values of six of these coils at the temperature at which they 
were stated by Hockin to be correct. The B.A. unit of one period is not, therefore, 
necessarily the same as that of another period. Every precaution was, of course, 
taken to ensure constancy, but with wire standards of resistance great difficulty is 
experienced. In after years # it proved possible to trace the changes in these coils 

* ' B.A. Elec. Stands. Committee Keport/ 1908. 



32 MR F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

with what appears to be a fair measure of success, and the corrections due to changes 
in the coils can in certain cases be calculated. 

It follows, therefore, that if the B.A. unit of any particular period is known in 
terms of the resistance of a column of mercury, and if the coils used for the absolute 
measurements remained constant in resistance from the time of their measurements 
in terms of a mercury column to the time of their absolute measurement, the results 
given in Table I. can in all cases be reduced to give the length of the column of 
mercury having a resistance of 1 ohm. 

But it is very probable that the resistance of many of the coils did not keep 
constant, and it is not possible for us to reduce the results except in a few cases. 
The instances referred to are the determinations by Lord Rayleigh, by Dr. Glaze- 
brook, by Viriamu Jones, and by Ayrton and Jones. In all of these cases 
comparisons were made with the B.A. standard coils, and the details of these 
comparisons have been preserved and published. In the Report of the B.A. Electrical 
Standards Committee for 1908, the changes in resistance of the coils used by Lord 
Rayleigh and Dr. Glazebrook have been traced, and a comparison of the mercury 
standards of resistance made by Lord Rayleigh, Dr. Glazebrook, and F. E. Smith 
is given in Table VIII. of the same report. This comparison, together with the notes 
on the standards used, enables us to express Lord Rayleigh's and Dr. Glazebrook's 
results in terms of the present mercury standards of the National Physical Laboratory, 
This we have done, the results being marked (S), while (A) indicates the results given 
by the author. 

Referring first to Lord Rayleigh's determination in 1882, we find that 
comparisons were made with certain B.A. unit coils and with mercury standards of 
resistance. However, the terminals of the latter were not at 0° C. but between 5° C. 
and 6° C., and it was shown by Dr. Glazebrook in 1888 that an error of 24 parts 
in 100,000 was introduced because of this. If we apply a correction of this 
amount, Lord Rayleigh's 1882 value of the ohm in centimetres of mercury becomes 
106*24 (1 4-0-00024) = 106*26 5 , and the 1883 value becomes 106'214 (l + 0*00024) 
= 106*239. These values are given in Table I. (within 1 part in 10,000) as 106*26 
and 106*24. 

Dr. Glazebrook's determination of the ohm was made in 1882, and he constructed 
mercury standards of resistance in 1888. The principal resistance coil employed in 
1882 was a platinum-silver coil known as " flat," and this also was used in 1888. In 
the interval it was assumed to have kept constant — there was at that time no certain 
evidence to the contrary. A careful survey of the history of the coils, which is 
published in the B.A. Beport for 1908, shows, however, that "flat" increased in 
resistance in the interval 1882-1888 by 41 parts in 100,000. Dr. Glazebrook^ 
value for the ohm in centimetres of mercury is 106*29, and this, when corrected for 
the change in the resistance coils, becomes 106*29 (1 — 0*00041) = 106*25. 

The coils used by Viriamu Jones in 1894 were compared with the B.A. standards 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 33 

by Dr. Glazebrook. One of these coils was No. 3715, and its value in 1894 was 
stated by Dr. Glazebrook to be r00026 ohms (international) at 14°'95 C, the 
relation between the B.A. unit and the ohm (international) being taken as 
1 ohm = 1*01358 B.A. unit. # From the results obtained at the N.P.L. in 1908 we 
conclude that this coil increased in resistance in the interval 1894-1908 by 7 parts in 
100,000. Its value in 1908 was measured to be 1*00066 international ohms at 
16°'0 C. or 1*00034 international ohms at 14°*95 C When allowance is made for the 
rise in resistance of 7 parts in 100,000, it will be seen that the difference from 
Dr. Glazebrook's value is 1 part in 100,000. We conclude, therefore, that the value 
given by Viriamu Jones in 1894 is not in error because of any uncertainty in the 
values of the resistance coils used. 

Similarly we have investigated the coils used by Ayrton and Jones in 1897, and 
we find the values in Board of Trade ohms agree with the values in international ohms 
within 1 or 2 parts in 100,000.. We conclude, therefore, that the length of the column 

-I A/? • Q AA 

of mercury representing the ohm is (from their experiments) — ; = 106*274 cm. 

Unfortunately, we are not sufficiently acquainted with the standards used by other 
investigators to reduce their results, and in the last column of the table the results 
given, except for the cases already dealt with, are those only in which mercury 
standards were available. 

Section 1. — Introductory. 

The instrument described in this paper is the outcome of a desire expressed by the 
late Prof. J. Viriamu Jones at a meeting of the British Association Committee on 
Practical Electrical Units and Standards, in 1893. Prof. Jones expressed the hope 
that in the near future there might be constructed an apparatus based on the method 
used by Lorenz, which would be kept in constant use in a national laboratory 
and embody in concrete form a proper ultimate standard of electrical resistance. 

In 1900 the Drapers Company of London promised to Prof. Jones the funds for 
the construction of such an instrument, and after Prof. Jones's death in 1901 the 
Company placed £700 at the disposal of the Executive Committee of the National 
Physical Laboratory in order that the instrument might be made. 

The apparatus was to be in memory of Prof. Jones, and to be constructed under 
the superintendence of the late Prof. Ayrton and of Dr. Glazebrook. Delay in 
proceeding with the work arose owing to the construction of the Ayrton-Jones 
current balance, and it was not until after completion of the balance in 1907 that a 
start was made. Unfortunately, Prof. Ayrton was in very poor health, and 
although keenly interested in the work he did not live to take any part in it. 

The form of apparatus eventually decided on was considerably larger than 
anticipated in 1893. The metal work was much too heavy for machining in the 

* ' B.A. Elec. Stands. Committee Reports/ 1892 and 1894. 
VOL. COXIV. — A. F 



34 



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RESISTANCE BY A METHOD BASED ON THAT OP JLOKENZ. 35 

Laboratory workshops, and Sir Andrew Noble, F.R.S., was approached with a view 
to this part of the instrument being made in the workshops of Sir W. G. Armstrong, 
Whitworth & Co. at Elswick. Sir Andrew Noble not only undertook that this 
should be done, but took such a keen interest in the work that he generously provided 
it at very much less than the cost price. 

In preparing the plans we had full access to the working drawings of the McGill- 
Lorenz apparatus made for Prof. Callendar by Messrs. Nalder Bros. & Co. 
Although we were unable to take much advantage of this opportunity owing to the 
big difference in the designs of the two instruments, we desire to express our thanks 
to Messrs. Nalder Bros, for their kindness. 

The instrument herein described possesses many new features of importance. It is 
permanent, but determinations of resistance made from time to time will not assume 
constancy of any dimensions ; it is comparatively easy to use, and the results obtained 
are believed to be of high precision. 

Section 2. — The Method of Lorenz. 

In this method, which was first employed by Lorenz^ in 1873, a rotating disc is 
placed in a magnetic field produced by a current which circulates through a coil 
coaxial with the disc. In the apparatus used by Lorenz wire brushes made contact 
with the disc at its circumference and centre, and the circuit was closed by connecting 
the brushes, through a galvanometer, with the extremities of a resistance R. The 
voltage induced by the uniform rotation of the disc at n revolutions per second is 
Mm, where M is the mutual inductance of the coil and a circle coincident with the 
edge of the disc, and i is the current through the coil. This induced voltage is 
balanced against the difference of potential Bi between the extremities of the 
resistance R through which the same current flows as through the coil. When 
equality of voltage is indicated by the galvanometer R = nM.. M is calculated from 
the dimensions of the coil and disc and thus R is found. The coil used was of a large 
number of turns in order to make M as great as possible. 

In the 'Philosophical Magazine' for November, 1882, Lord Rayleigh compares 
the method of Lorenz with other methods. After stating that he is disposed to 
consider Lorenz's method the best, he proceeds to deal with some of the difficulties. 
The first of these is the smallness of the resistance R which can be directly measured, 
and this led Lord Rayleigh to adopt a system of shunted resistances which, for part 
of our work, we also have used. Lord Rayleigh remarks that the influence of 
terrestrial magnetism and the thermo-electric effects at the sliding contacts are both 
very appreciable and give rise to trouble during the observations, but they can be 
eliminated by observing only the effect of reversing the battery current. 

The more important portion of Lord Rayleigh's comments deals with the ratio of 

* Lorenz, < Pogg. Ann.,' 149, p. 251, 1873. 

E 2 



36 ME. F. E, SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

the dimensions of the coil and disc. The results are given of some calculations of the 
mutual inductance M of a coil of radius A and circle of radius a. The rates of 
variation of M with change in the diameter of the coil are calculated for various 
values of a/A, and it is shown that when a/A increases in value the rate of variation 
of M with change in a and also of A increases. Further, it is shown that, by using 
two coils separated to a considerable distance, it is possible so to proportion the radii 
and the distance apart of the coils that the error of mean radius of the coil does not 
affect the result ; the diameter of the disc and the distance apart of the two coils are 
then the important quantities. Lord Rayleigh remarks : " It is clear that M 
vanishes both when A is very small and when it is very large ; from which it follows 
that there must be some value of A for which the effect is a maximum and therefore 
independent of small variations of A." The same is, of course, true for the disc ; by 
suitably proportioning the dimensions, the error of mean radius of the disc may be 
rendered negligibly small. This fact led Mr. A. Oampbell # to design a standard of 
mutual inductance in which the radius of the secondary is not required to be known 
with great precision ; it also guided us in fixing the dimensions of our coils and discs 
so that the diameters of the latter need not be accurately known. 

In the first and second series of experiments carried out by Lord Rayleigh the 
inductance coils were situated nearly in the plane of the revolving disc as in Lorenz's 
original use of the method. In the third series the coils were separated from the 
disc to such a distance as to render the accuracy of the results practically independent 
of the mean radius of the coils. It is right to say here that in the design of the 
apparatus described in this paper we were largely influenced by Lord Rayleigh's 
investigations and by Mr. Campbell's work. 

Of the experimental difficulties noted by Lord Rayleigh the more important are : 
(l) troublesome thermo-electric effects at the sliding contacts notwithstanding that 
the edge of the disc was amalgamated with mercury, and (2) effects due to terrestrial 
magnetism. As before stated, both of these effects are eliminated by taking a 
sufficient number of readings with reversals of the current, but it is evident that good 
readings cannot be taken if the magnitude of the effects is subject to sudden 
fluctuation. 

In 1890 Prof. J. V. Jones t made a number of suggestions towards a determination 
of the ohm. He had made a number of experiments in his laboratory at the 
University College, Cardiff, and stated that he was of opinion that, if apparatus were 
constructed on a large scale and with a certain perfecting of detail, a single set of 
observations would give a result accurate within 1 part in 10,000. In the electrical 
observations the principal difficulties which he had contended with were: (l) varia- 
tions in the thermo-electric effects at the brush contacts, and (2) variations in the 
rate of rotation of the disc. A considerable reduction of the first difficulty was 

* Campbell, ' Roy. Soc. Proc./ A, vol. 79, p. 428. 
t Paper read before the British Association, 1890. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 37 

brought about by using a brush consisting of a single wire perforated by a channel 
through which a constant flow of mercury was maintained. With regard to the 
variation of speed, Prof. Jones was of opinion that " no time or trouble spent in 
securing a constant speed will be lost for the purpose in view." It was suggested 
that the standard coil should consist of a single layer of wire, as the mutual 
inductance of the coils and disc could then be calculated with great accuracy. 

The apparatus made by Messrs. Nalder Bros. # for Prof. Callendar, who was then 
at the McGill University, Montreal, was, in general arrangement and dimensions, 
similar to Prof. Jones's Cardiff apparatus. Contact with the edge of the disc was 
made by three small tangential phosphor-bronze tubes lightly pressed on it, at points 
separated by angular distances of 120 degrees, and through these tubes mercury 
flowed on to the edge of the disc. The employment of three brushes was suggested 
by Rowland to eliminate small errors due to imperfect centering of the coil and disc, 
and was a distinct improvement. The possible sources of error were considered, and 
in a paper by Prof. Ayrton and Prof. Jones # an equation is given showing the rates 
of variation of the mutual inductance of the coil and disc with changes (l) in the 
radius A of the coil, (2) in the radius a of the disc, and (3) in the axial length 2x of 
the helix. The equation given is 

^= 1-246 ^+2-346 — -0-0997— . 

M A a x 

The value of M was 45814*5 cm., of A 267039 cm., and of a 16*5354 cm., from which 
it is readily calculated that an error of 0'002 cm. in the measurement of the diameter 
of the disc introduced an error of nearly 5 parts in 100,000 in the value of M. An 
error in the measurement of the diameter of the coil of 0*002 cm. introduced an error 
of 14 parts in 100,000 in the value of M. 

It will be seen that the experimental difficulties experienced by observers using the 
Lorenz method were mainly due to thermo-electric troubles at the brush contacts, 
want of uniformity of speed, and the effect of terrestrial magnetism. There are no 
difficulties attendant on the accurate evaluation of the mutual inductance of a coil 
and disc if the dimensions are accurately known. If a single layer coil is used, its 
dimensions may be determined with great accuracy ; if a coil of many layers is 
employed, it appears best to follow Lord Rayleigh's practice and use two coils at a 
considerable distance apart in order that errors of measurement of the radius of a coil 
may be rendered negligibly small. 

The Lorenz apparatus described in this paper was designed in 1908, our object being 
to determine a resistance in absolute- measure with a precision within a few parts in 
a hundred thousand. The apparatus described eliminates the effect of terrestrial 
magnetism and largely reduces thermo-electric troubles at the brush contacts by the 
employment of two discs of equal diameters. The coils are of one layer only and 



* i 



British Association Reports/ 1897. 



38 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



their dimensions, together with those of the discs, may be determined with ease and 
with precision ; in addition, any possible error in the measurement of the diameters of 
the discs is made negligibly small. Instead of employing one rotating conductor, a 
number are used, and the induced voltages may be placed in parallel or in series. The 
machine is of the best construction, great attention being paid to lubrication in order 
to avoid irregularities in the speed. As already stated, the heavy metal work was 
kindly undertaken by Sir W. G. Armstrong, Whitworth & Co,, but the smaller work 
such as the chronograph, commutators, brushes, coil fittings, &c, many of which 
demanded great skill in their construction, were made by Mr. F. H. Murfitt, who 
has charge of the instrument shop of the Laboratory, and whom we also thank for 
many suggestions, 

Section 3. — General Description. 

The instrument consists of two metallic discs which support ten conducting wires 
rotating in a magnetic field produced by a current in four coils ; an electric motor is 
used as the source of power. Phosphor-bronze wire brushes make contact with 
segments made of the same alloy attached to the ends of the rotating wires, and the 
difference of potential between the brush contacts at the ends of a single wire, or that 
between five wires, is balanced against the difference of potential between two points 
on a standard resistance, the current through which is the same as that flowing through 
the four coils, A diagrammatic sketch of the arrangement is shown in fig. 1. 




Fig. 1. 

It will be seen that the current in the coils 1 and 2 is in the opposite direction to 
that in the coils 3 and 4. The resulting magnetic fields are opposed in direction, and 
the value of the field at points in the neighbourhood of the edge of a disc is zero 
or nearly so. A difference of potential is produced between the ends of a rotating 
conductor, and its value is dependent on the position only of the ends of the 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 39 

conductor and not upon its shape, conditionally that the conductor passes through the 
coils carrying the current. Thus the difference of potential at the extremities of a 
conductor ACDB is not altered if its shape is changed to AEFB. The segments 
which form the ends of the conductors are insulated from each other and from the 
discs, 

The brushes consist of phosphor-bronze wires stretched by two spiral springs, and 
resemble violin bows. Each brush makes contact with one or two segments over a 
length varying from 5 to 6 cm., and leaves a segment at a tangent thus making the 
pressure greatest at the mid-point of contact. Petrol is employed as a lubricant. 

There are two principal ways of using the apparatus. In the first the ten brushes 
are included in a circuit so as to be in series. When each brush is in contact with a 
single segment, the differences of potential due to five rotating conductors are added 
together, the remaining five conductors being ineffective. When each brush connects 
two neighbouring segments the ten rotating conductors are arranged in five sets of 
two in parallel and the total potential difference is the same as before. It is easily 
seen that by having a comparatively large arc of contact between each brush and a 
segment (or segments) and twice as many segments as brushes, the circuit made 
through the brush contacts is never broken. 

In the second method, the brushes are divided into two sets of five in parallel and 
the total potential difference is the same as that of a single rotating conductor. 

All the coils are wound with bare copper wire on hollow marble cylinders having 
double-threaded screw grooves cut on the surfaces. The two wires on any one 
cylinder form two adjacent helices which may be connected in series or in parallel. 
In the general use of the instrument they were connected in series, but they may at 
any time be disconnected from each other and an insulation test made between them. 
There are eight helices in all and these are connected by means of small concentric 
cables to a plug board and commutators which enable the direction of the current in 
any coil to be changed at will. 

Each of the cylinders is mounted on a strong metal support and its position with 
regard to a disc may be altered with ease by screw adjustments. The distance 
between the mid-planes of two coils is measured by means of microscopes. 

The electric motor used for driving is situated at a considerable distance from any 
one of the coils, and its influence on the result was calculated and also experimentally 
proved to be negligibly small. A commutator is fixed to the axle of the motor, and 
this serves to charge and discharge a condenser placed in one arm of a Wheatstone 
bridge. By keeping the bridge permanently balanced, the speed of the Lorenz 
apparatus is maintained constant. A directly driven chronograph enables the speed 
to be calculated. 

The whole instrument is supported on gun-metal rails embedded in a solid block 
made with Keene's cement, sand, and ballast. With the exception of the motor, no 
magnetic material is used in the construction of the apparatus or supports. The 



40 



ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



complete apparatus is diagrammatically shown in fig. 2 ; fig. 3 is a photograph of the 
instrument taken from the end farther from the motor. The overall length is seven 
metres. 

ivi w o , o 



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^c:.'.„.„ 




™.-/ 



r^fix. 




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i 




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Fig. 2. 
M, motor ; W, fly-wheel ; DD, discs ; CCCC, coils on cylinders. 

Section 4. — The Motor, and Fly-wheel. 

The electric motor M, fig. 2, is coupled to a shaft supporting a fly-wheel W; the 
fly-wheel shaft is in turn coupled to another supporting one of the discs, and the latter 
shaft is coupled to a similar one supporting the other disc. The coupling between the 
fly-wheel shaft and that supporting one of the discs is an insulating one, washers and 
tubes of stabilit being used to secure good insulation. 

The electric motor is a shunt- wound one supplied by Messrs Crompton & Co., Ltd. 
The machine has four poles and has a commutator of the radial type. The shaft is of 
phosphor bronze and the motor is mounted on a base of the same alloy in order to 
reduce the quantity of magnetic material to a minimum. The stray field of the 
motor in an axial direction is surprisingly small, its intensity at a distance of 400 cm. 
amounting to 0*0006 C.G.S. units only. Our original intention was to enclose the 
motor in a double shell of soft iron, and the base of such a shell is interposed between 
the motor and its support; the results of our observations on the stray field showed 
this to be unnecessary and the idea was abandoned. The effect of the motor on the 
mutual inductance of the coils and discs is discussed in Section 19. 

The fly-wheel is of phosphor bronze. The outer diameter is 50 cm. and the weight 
of metal in the rim is about 80 kgr. 



Section 5. — The Rotating Discs and Their Supports. 

The portion to the right of the instrument (fig. 2) consists of two similar parts, and 
only one of these will be described. Fig. 4 shows one of the parts. 

The disc D is of rolled phosphor bronze fitted on a shaft S made of an alloy of 
copper and aluminium (10 per cent, aluminium and 90 per cent, copper). The 
original intention was to have the shaft of phosphor bronze, and two such shafts were 
made but were rejected on account of their appreciable magnetic susceptibility. 
Phosphor-bronze billets having the requisite magnetic properties could readily be 
obtained, but in such cases they failed to give satisfactory results with the mechanical 
tests and were therefore unsuited for shafting. The diameter of a disc is about 53 cm., 
and that of the shaft is 5 cm. 



F. E. Smith 



Phil. Trans., A, vol. 214. 




Fig. 3. 




Fig. 4. 



To face p. 40. 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 41 

The shaft runs in bearings made of a special alloy of tin (69 per cent.), zinc (29*6 
per cent.), and copper (1*4 per cent.), this being the best non-magnetic alloy we know 
of in which a copper aluminium shaft will run without trouble. The pedestals PP and 
bed B, supporting the bearings, are of phosphor bronze, except for slabs of stabilit 
T, 6 mm. in thickness, which insulate the bearings from the bed. The upper part of 
a pedestal is bolted to the lower part by phosphor-bronze bolts L, stabilit washers 
W, and stabilit tubes being used to insulate the upper and lower parts. The insulation 
resistance between the shaft and the bed has been measured on many occasions, but 
has never been less than a thousand megohms. If two metallic discs with uninsulated 
rims and conductors are employed, such insulation is essential, but it is not necessary 
with insulated conductors and segments such as we have used. 

The oil feed (F) and return pipes (R) are of copper and are insulated from the 
pedestal by glass and ebonite tubes and oil-resisting rubber washers. Oil throwers 
are fitted to the shaft and effectually prevent the oil travelling outwards from the 
bearings. 

Section 6. — The Rotating Conductors which Pass from the Edge of One 

Disc to the Edge of the Other Disc. 

We have already briefly described the system of ten conducting wires passing from 
disc to disc, and it will be realised that the discs in the present apparatus serve only 
to support the radial conductors and are employed for no other purpose. The wires 
used consist of No. 26 double silk-covered copper wire, shellacked, and covered with 
silk tube. 

In order to place the ten wires in position, the shaft was drilled centrally and 
parallel to its length from the coupling between the two discs to points within 20 cm. 
of the discs, and radial holes were drilled in the shaft in these latter positions. On 
the coupling between the discs blocks of ebonite are screwed, and these support 
terminals to which the ends of the wires can be attached. Ten wires pass from the 
segments attached to the edge of one disc, through a channel milled in the side of the 
disc, and again through a brass tube screwed on the shaft, into a radial hole ; after 
passing through the central hole drilled in the shaft they emerge at the coupling, and 
the free ends of the wires are attached to terminals on the ebonite blocks. Similarly, 
ten wires pass from the edge of the other disc to corresponding terminals, and by 
making suitable connections between the terminals we obtain a system of ten 
conductors which pass between the segments on the two discs. As already explained, 
the path of the conductors is unimportant, conditionally that they pass through the 
coils. 

On the edge of a disc ten segments of stabilit are screwed. Some of these can be 
seen in the photograph (fig. 3), and in the sketch (fig. 5) ; a sectional view of a 
segment attached to a disc is shown in fig. 5c. On emerging from the channel milled 

VOL. CCXIV. — A. G 



42 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



in the disc, the wires pass along a groove G, milled in the stability and are soldered to 
ten phosphor-bronze wire segments, of square section, which are screwed to the stabilit 
in the manner shown in the illustration. This method was adopted because of the 
strain to which the segments are subjected when the machine is running at full speed 
and which results in an increase in the effective diameter of a disc, measurements of 
which are given in Section 16. The wire for the phosphor-bronze segments is 25 sq. mm. 
in cross section and was kindly drawn for us by the London Electric Wire Company. 
Some of it was further drawn down into circular section wire of 0*12 mm. diameter, 
and this latter was employed for the brushes. Phosphor bronze was chosen for the 
segments and brushes because of the negligible tendency to bind and tear when this 
alloy is employed for moving parts in contact. 




Fig. 5. 

Fig. 5 shows five segments in position on a disc and illustrates the manner in which 
the wires pass from the segments to the main shaft. Before making any resistance 
measurements the shaft was rotated slowly, and with the aid of a small portable 
milling machine a smooth surface was turned on the segments. The sharp edges 
which resulted at e, e (fig. 5) were rounded in order to avoid a cutting action on the 
wire brushes. 



Section 7. — The Brushes and Their Lubrication. 

To obtain experience regarding the best form and number of brushes, preliminary 
apparatus consisting of two rotating discs of phosphor bronze was constructed in 1908. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 43 

The apparatus was astatic and our experiments were directed to eliminating trouble 
due to thermo-electric effects at the brush contacts. 

The first brushes experimented with were of fine phosphor-bronze wire, each brush 
consisting of about 50 wires bound together after the manner of a common pencil 
brush. A light spiral spring ensured contact between a brush and the edge of a disc. 
A circuit through brushes, discs, and a galvanometer was completed, and the discs 
(25 cm, in diameter) were run at a speed of about 1500 revolutions per minute. 
Whatever thermo-electric effects existed at the points of contact of the brushes 
on one disc must have been in opposition to those at the brush contacts on the 
other disc, but the galvanometer deflection was far from steady, the variations in the 
thermo-electric effects amounting to about 0*0001 volt. Although these preliminary- 
experiments of ours were so unsatisfactory, the results were much better than those 
obtained when the circuit was completed through brush contacts at the edge of one 
disc and others on the shaft of the apparatus. 

An even more disturbing feature of our experiments was the inconstancy of the 
resistance of the galvanometer circuit. With one brush on each disc the resistance of 
the contacts was measured as something less than 01 ohm when the discs were 
stationary, but when the discs were rotating the contact resistance increased to 5, 10 
and 20 ohms, and at times the circuit appeared to be broken. This effect was 
undoubtedly due to a vibration of the brushes brought about by the motion of the air 
at the edges of the discs. In practice it was not possible to prevent this by increasing 
the pressure applied to a brush, as by doing so the discs were worn away very rapidly 
and the points of contact became very hot. Shielding devices were tried with some 
small success, but the most favourable results were obtained by employing a number 
of brushes in parallel and placing in series with each brush a resistance of 1 ohms, 
thus ensuring that any increase or diminution of the resistance of one brush contact 
would have but little effect on the circuit as a whole. Our experimental results 
showed clearly that the employment of a large number of brushes was beneficial, but 
an increase in the number did not greatly improve the results, unless a resistance was 
placed in series with each before the brushes associated with any one disc were placed 
in parallel. 

A further advance was made by lubricating the edges of the discs. Fatty oils and 
greases are impossible for such a purpose as the contact resistances are enormously 
increased and the general result is far worse than when no lubricant at all is used. 
Acheson graphite, aquadag and oildag were tried, but the results were not satis- 
factory. It is not very well known that paraffin oil improves all ordinary contacts 
such as those associated with slide wires, plugs, &c, but throughout the National 
Physical Laboratory paraffin oil is largely used for such purposes. We tried it there- 
fore as a lubricant for the brushes. There was a marked improvement in the results, 
and we continued to use it in this preliminary work. Many kinds of brushes were 
tried : some were wires of phosphor bronze, some of phosphor-bronze gauze, some of 

G 2 



44 



MR* F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



copper wire, and solid slipper brushes of phosphor bronze were also used. The most 
satisfactory results were obtained with brushes of fine wire. 

After the erection of the Lorenz apparatus we continued our experiments with 
brushes. The discs of the new apparatus are about 53 cm. in diameter and they 
revolve about 1050 times per minute, so that the velocity at the edge of a disc is 
about 1750 m. or 1 mile per minute. This velocity is greater than the velocity at 
the edges of the discs in our preliminary apparatus, and we found the brush difficulties 
correspondingly increased. The variation in the resistance of the galvanometer 
circuit due to the varying contact resistances rendered accurate work impossible, and 
we were led to design a brush which would not be set in vibration except to a very 
small extent and by the use of which thermo-electric effects would produce compara- 
tively small disturbances. A short description of these brushes appeared in the 
'Annual Eeport of the National Physical Laboratory for 1911/ and we are aware 
that the short description there published has led several investigators to try similar 
brushes, and the results have been reported as satisfactory. 

We were guided in the design by the following considerations : ( 1 ) a fine cylindrical 
wire if stretched offers little resistance to a stream of air, and whatever vibration is 
set up will probably be regular and can be controlled by adjustment of the tension on 
the wire ; (2) if a wire such as AB (fig. 6) is in tension and makes contact with the 




Fig. 6. 



edge of a rotating disc D over the arc EOF, the pressure will be greatest at a point 
close to C the mid-point, and will gradually fall in value until at E and F it is zero. 
The maximum rise of temperature and the maximum thermo-electric effect will 
probably be at 0, and the thermo-electric force will gradually diminish as we approach 
E and F and be zero at these points. Without the completion of any other circuit it 
is clear that electric currents will be produced which will flow from the wire to the 
disc in some parts and from the disc to the wire in other parts. If, therefore, a 
galvanometer is included in the circuit containing the junction of wire and disc, the 
resultant deflection will be very much less than that which would be produced by the 
maximum thermo-electric effect. A number of stretched wires in parallel should, of 
course, give better results than one. The form of brush actually used in our 
experiments is depicted in fig. 6. The size was largely governed by the arc of contact 
desired and the number of segments on the discs. 



KESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 45 

The brushes are of phosphor-bronze wire 0*12 mm. in diameter, and to obtain as 
small a thermo-electric effect at the contacts as possible the wire was drawn from 
other wire of square section similar to that employed for the segments on the discs. 
To make a brush, the fine wire is wound in screw cut grooves of 0*25 mm. pitch, cut 
on small brass cylinders which can rotate about, or be clamped to, the axles AA'. 
The ends of the wire are soldered to the cylinders and the wire brush thus formed is 
put in tension by operating the milled heads HH. The spiral springs SS' are of stout 
phosphor-bronze wire and are soldered to brass rods RR/, of square section, which 
slide in square cut grooves cut in the frame F. The overall length of a brush is 
20 cm. In practice we found the tension required for good working to be very small ; 
a tension corresponding to the pull produced by the suspension of a 200 gr. weight 
was found to be most satisfactory. Some of the brushes were made with eight 
turns of wire, some with three turns, and a few with one turn ; the majority were of 
three turns. 

From the moment we commenced to use this style of brush with petrol as a 
lubricant, the thermo-electric variations produced very much less trouble. The 
variations are not more than one-fiftieth of those found with any other form of brush 
we have experimented with, and the variation in the resistance of the contacts when 
ten brushes are placed in series is so small that the sensitiveness of the galvanometer 
remains constant within the limits of error of our measurements. However, in the 
case of ten brushes in series, the tension on the wires has been somewhat greater than 
that recorded above. 

Without a lubricant the brushes are not satisfactory. The wire is gripped by the 
disc and under certain conditions the brushes vibrate longitudinally and produce at 
the same time a chattering noise. Also the wire and disc become rugged and the 
temperature at the point of contact is very much greater than when petrol is used as 
a lubricant. We believe the petrol to have three beneficial influences : (l) as a 
lubricant ; (2) as a cooling agent ; (3) as a cleanser getting rid of all traces of grease and 
dirt. The amount of petrol to be supplied and the manner of supply was the subject of 
many experiments, but it is sufficient to state here the manner eventually adopted. 

Fig. 5 shows a brush in position and a petrol supply vessel V. The brush is 
secured to a massive phosphor-bronze ring R (see figs. 3 and 5) by the screws SS', 
and it can be placed in contact with a segment or removed from such contact by 
loosening the terminal head H and giving the brush frame a circular motion about the 
pivot screw S'. The screws SS 7 and the brass distance pieces BB 7 are insulated from 
the ring R by means of ebonite sleeves, the latter allowing of some adjustment of the 
screws SS'. The ring R is supported by the bed B (fig. 4) but is insulated from it. 
Stretched phosphor-bronze wires support the ring laterally and give it the necessary 
rigidity. 

The petrol supply vessel is of glass, the tube being drawn down to a capillary about 
0*5 mm. in diameter at its lower end. On the narrow glass tube thus formed, a 



46 MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

piece of circular lamp wick about 3 cm. long is slipped ; the lower end of this wick 
just touches the rotating segments and thus feeds petrol directly on to the surface of 
the segments. The usual rate of supply of petrol was about 500 c.c. in twenty 
minutes. When the apparatus is running well, an observer may place the end of a 
finger on the rotating segments and find it well flooded with petrol removed from the 
rim. The wicks are renewed at least every day and the brushes wiped with clean 
chamois leather. In our experiments a set of ten brushes lasted usually for six or 
nine complete sets of observations ; after that number the surfaces of the wires 
became somewhat rough and the results were not quite so satisfactory. The 
magnitude of the changes of the thermo-electric effects with two sets of five brushes 
in parallel will be realized when it is said that often for intervals of twenty minutes 
the rapid variations in the total thermo-electric voltage did not exceed 0*1 microvolt. 
A slow progressive variation was commonly observed, but this was not a source of 
trouble. 

Section 8.— The Coil Supports. 

Each marble cylinder weighs about 50 kgr. and is supported on a phosphor-bronze 
cradle C (fig. 4) so that its axis is coincident with the axis of the shaft. The base of 
the support is a triangular casting G, which is supported in turn by three levelling 
screws A on the " hole, slot, and plane " principle. The pitch of the levelling screws 
is one millimetre and the heads of the screws are divided into one hundred equal 
divisions, thus enabling any particular vertical motion to be repeated within one- 
hundredth of a millimetre. 

A second triangular casting K carries the levelling screws and is fitted over a large 
central stud attached to a slide, so that the cylinder may be rotated about a vertical 
axis if necessary. The maximum angular motion is 12 degrees and the magnitude of 
any motion can be directly read on the engraved head of the horizontal screw E to 
half a minute of arc. Backlash is avoided by the use of strong phosphor-bronze 
springs. 

A cylinder and the cradle supporting it can be moved 5 cm. in two horizontal 
directions at right angles by means of two slides, the motions being controlled by screws, 
the heads H of which are divided to read hundredths of a millimetre as in the case of 
the screws for the vertical motion. For these movements also strong phosphor-bronze 
springs are employed to prevent backlash. 

Section 9. — Lubrication of Bearings. 

The bearings of the machine are nine in number, each of those of the motor being 
about 9 cm. long and each of the remaining seven about 14'5 cm. long. The lubricant 
used is best turbine oil which is fed to the bearings under a pressure of about 
15 lbs. per square inch, the rate of supply being a cubic foot of oil every five minutes. 
The oil supply tank and pump are of phosphor bronze and are situated under the 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 47 

floor, l|- m. below the level of the bearings of the main shaft, and 4 m. distant 
from the nearest point of the machine. The pump is driven by a small motor of one- 
eighth horse-power and forces the oil through copper pipes to the bearings of the 
machine. After passing through the bearings the oil returns through copper drain 
pipes to the tank and is strained through fine copper gauze before again entering the 
pump. The system works excellently, no trouble whatever having been experienced. 

Section 10. — Regulation of Speed. 

On the axle of the motor a commutator is fitted which serves to charge and discharge 
a condenser four times for each revolution of the shaft. The condenser is placed in 
one arm of a Wheatstone bridge, the other arms of which are platinum-silver resistance 
coils. Balance of the bridge results for a particular frequency only of charge and 
discharge, and to maintain a balance over a considerable length of time the frequency 
and therefore the speed of the Lorenz apparatus must be kept constant. To ensure 
constancy of the arms of the bridge, the condenser and the platinum- silver resistances 
were kept in a constant temperature room ; a small variable resistance in series with 
one of the arms was in general adjusted to secure a balance when the speed was that 
best suited for the resistance measurements, but after the latter measurements had 
commenced the speed only was controlled to maintain the balance. The galvanometer 
used was a suspended coil instrument, the spot from which was received on a ground 
glass scale mounted over the fly-wheel ; an assistant observer, S. Watts, applied a 
variable pressure to the fly-wheel and so maintained the balance of the bridge. A 
change in the speed of the Lorenz apparatus of 1 part in 10,000 produced a deflection 
of the galvanometer spot of 4 mm., and in general, a balance was maintained for twenty 
minutes or more with a maximum deflection not greater than 2 mm. ; occasionally, 
better results than this were obtained. An adjustable resistance is in series with the 
field coils of the motor, and before attempting to govern the speed this resistance is 
altered until a speed results which is very slightly greater than that desired. 

The motor was run from a battery of large storage cells and very good results were 
obtained with no regulation at all, if the current flowed sufficiently long (generally 
from one to two hours) to raise the field coils to an approximately constant tempe- 
rature. However, the method finally adopted enabled resistance measurements to be 
made a few minutes after starting the motor. A somewhat similar method of 
controlling speed for measurements of capacity has been in use for several years both 
at the National Physical Laboratory and at the Bureau of Standards. 

Section 1 1 . — Method oe Recording the Speed. 

The chronograph described is very similar to one designed at the Bureau of 
Standards, # and made by the Societe Genevoise, for details of which the writer 



* i 



Bull. Bureau of Standards/ vol. 3, p. 561. 



48 ME. F. E. SMITH ON THE ABSOLUTE MEASUKEMENTS OF A 

is indebted to Dr. Rosa and other members of the Bureau of Standards. The 
instrument described below, which is of brass, was made by Mr. Mubfitt of the 
Instrument Department of the Laboratory. 

The method of recording the speed is a direct one. A chronograph drum is geared 
to the main shaft of the apparatus through the medium of two worm wheels and two 
spur wheels, the gear ratio being 174. The usual speed of the main apparatus is 
about 1044 per minute, and under such conditions the drum of the chronograph makes 
one revolution in ten seconds. An electro-magnet is supported on a carriage which is 
connected to a split-nut engaging with a screw of 1 mm. pitch. The direction of 
travel of the carriage is parallel to the axis of the drum and it advances 1 mm. for 
each revolution of the drum. Every second a current passes through the electro- 
magnet and the latter operates a small punch, which, striking through a typewriting 
ribbon, prints a dot on a sheet of paper carried by the drum. The circumference of 
the drum is exactly 500 mm., so that when six revolutions per minute are made, 
successively recorded dots are 50 mm. apart. The split-nut fixed to the carriage 
carrying the electro-magnet can be disengaged from the driving screw and the carriage 
rapidly run along the rails supporting it. When the printing punch is pressed 
forward during this operation a line parallel to the direction of travel of the electro- 
magnet is printed on the paper. This line is hereafter called the base line and by 
measuring the angle between it and a row of dots recorded during a run, the speed 
can be calculated, with great accuracy. 

The method of calculating the speed is as follows : At a speed of exactly 1044 
revolutions per minute there are 1044/174 = 6'0000 revolutions of the chronograph 
drum per minute. The resulting record will therefore be ten rows of dots, the dots 
being 1 mm. apart, and the rows 50 mm. apart and parallel to the base line. If the 
speed is slightly diminished or increased, the rows of dots will slope upwards or 
downwards. In practice, a set of observations for the measurement of a resistance is 
made to last for at least 1000 seconds (i.e., about 17 minutes), and the resulting rows 
of dots are therefore more than 10 cm. long. From a portion of the base line lying 
beneath (or above) a row of dots a length of 10 cm. is marked off, and from the 
extremities of the 1 cm. line ordinates are drawn to the nearest row of dots. If these 
ordinates are equal in length the rows are parallel to the base line, and the distance 
between two dots recording an interval of 1000 seconds (10 cm. run) is 50,000 mm. if 
the distance is measured along the trail of the recorded seconds. If the ordinates are 
not equal in length the machine was running either faster or slower than 1044 
revolutions per minute ; which of these holds good is decided by the direction of the 
slope. Suppose the difference between the ordinates is 14*5 mm. and that the machine 
was running faster than 1044 per minute. The mean speed is calculated to be 

1044 x = 1044'30o revolutions per minute. It is clear that a difference 

50,000-14-5 F 

between the ordinates of half a millimetre corresponds to 1 part in 100,000 of the 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 49 

speed. The drum is 50 cm. long and a record lasting 80 minutes can therefore 
be made. 

Fig. 7 is a full-size reproduction of a portion of a record, taken June 26, 1913. The 
recorded speed is 1043'96 2 revolutions per minute. The rows of dots enable the mean 
speed to be calculated with an error certainly not greater than 1 part in 100,000, and 
the speed throughout (as illustrated by the rows of dots lying in a straight line and 
not a wavy one) is wonderfully uniform. The general fluctuations of speed cannot 
be detected on the record, but their magnitude has been estimated in another way as 
explained in Section 23. 

A mercury contact is used on the pendulum of the standard clock and a relay is 
operated continuously by the current which passes. The rate of the clock is of course 
allowed for. 

Section 12. — Magnetic Tests. 

Magnetic tests were made on all the materials employed. With the exception of 
the motor, we are satisfied that the permeability of no part of the machine and the 
concrete bed on which it rests differs from unity by more than 2 parts in 100,000. 

Every part of the machine bears a distinctive number. When each part was cast, 
a lug about 5 cm. long and 5 cm. in diameter was cast with it, and after being stamped 
with the same number as the casting it was forwarded by Sir W. G. Armstrong, 
Whitworth & Co. to be tested for magnetic quality. In the case of springs, rods, 
tubes, &c, pieces were cut from these and used for the tests. 

The method of testing was similar to that employed for the parts of the current 
balance. # Soft iron wire and ferrous sulphate were used to calibrate the apparatus 
and sufficient sensitiveness was obtained to detect a difference from unit permeability 
of about 1 part in 100,000. Thus, when powdered ferrous sulphate having a 
permeability of about 1*00044 was contained in a glass tube having the same cross 
section as most of the test pieces, the resulting galvanometer deflection was 5*2 mm. 
The test pieces of marble were of much larger cross section than the glass tube and 
the sensitiveness was correspondingly increased. 

Trouble was experienced with a number of brass rods and tubes and with the shaft 

of the apparatus. The first shaft was made of phosphor bronze and had a magnetic 

permeability of 1*006 ; in consequence this shaft was rejected. The material obtained 

for the second shaft was of an alloy of copper and aluminium (copper 90 per cent., 

aluminium 10 per cent.) and was also rejected as its permeability was about 1*002. 

The third shaft was made with specially pure aluminium and copper in the same 

proportions as before, and its permeability differs from that of air by an amount too 

small to be detected. The billet for the shaft was prepared by The Broughton Copper 

Company, Limited, of Manchester, and we thank them for the care taken in its 

preparation. 

* 'Phil. Trans/ A, vol. 207, p. 475, 1908. 

VOL. CCXIV.— A. H 



50 ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 






*£*•«%«• *««»f«*«tt»*«t*«tt+94******«e«t 



•••••"•••—♦•* •'•"••••"••••"•••••••••MMtMM«-..M«*M,»«M., 






*••" ' "•♦••• * " ' ••••..,., ....... .•».«.,«»»«••, 

Base Line. 






Fig. 7. 



EESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 51 

The samples of Portland cement tested varied in magnetic permeability from 1*0005 
to 1*0020. It is possible that metallic iron finds its way into the cement from the 
steel grinding machines, and the Associated Portland Cement Manufacturers very 
kindly offered to exclude such contamination by grinding some cement through mill- 
stones. At the same time the manufacturers pointed out that a certain amount of oxide 
of iron in combination with lime and silica is invariably present in Portland cement, 
and Keene's cement was suggested as being suitable for our purpose. This is a 
white cement absolutely free from iron compounds of any sort. We tested several 
samples for magnetic quality, and finding it quite satisfactory decided to use it for 
the block on which the apparatus rests. No trouble was experienced in obtaining 
sand and ballast free from magnetic substances. 

Section 13. — Construction and Measurement of the Coils. 

" First Statuary " Carrara marble was chosen for the material of the cylinders. We 
were guided in our choice by our experience with the cylinders for the Ayrton- Jones 
current balance, the tests then made showing marble to be an excellent electrical 
insulator and of negligible magnetic susceptibility. 

The cylinders were prepared in the rough by Messrs Walton, Gooddy & Cripps, whom 
we thank for the trouble they took in choosing masses of marble practically free from 
veins. The small shiny specks which are often present in Carrara marble consist of 
iron pyrites which has a magnetic susceptibility of about 0*0005. The conductivity 
of iron pyrites is much greater than that of marble and any small specks on the 
surface of a cylinder should therefore be removed ; we found that a small crystal of 
pyrites pressed between two plates of copper reduced the insulation between the 
plates from a value which was practically infinity to 1000 ohms. Clearly a large 
number of crystals at the surface of a marble cylinder would introduce a serious source 
of error. 

The coefficient of expansion of the marble was determined by direct measurements 
on the cylinders and found to be 5*0 x 10~ 6 for 1° C, the temperature range being 
from 11° C. to 20° C. These measurements are described in the section dealing with 
the measurements of the mean diameters of the coils. 

The cylinders chosen are free from flaws and cavities. Of six cylinders which were 
submitted, two exhibited " ground flaws " and were in consequence rejected. These 
"ground flaws" appear to have been produced by a rupture in the marble many 
thousands of years ago and subsequent re-union by pressure or equivalent agency. 
In all, Messrs Walton, Gooddy & Cripps prepared 16 cylinders in the rough and the 
four best of them were chosen for our work. 

The marble cylinders for the Ayrton-Jones current balance were baked in an oven 
at 140° C. for 30 hours and afterwards immersed in hot paraffin wax. At the same 
time a marble rod was subjected to similar treatment, and since that time this rod has 

H 2 



52 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



been kept under observation in the Metrology Department. Possibly there is a flaw 
in the rod, but it is certain its length has not kept so constant as other rods which 
were not baked and immersed in wax. We decided, therefore, to omit the treatment 
in the case of the four cylinders for the Lorenz apparatus. 

The cylinders were turned in a manner very similar to that employed for the 
suspended coils of the current balance. The inner and end surfaces of a cylinder 
were turned with the cylinder fixed to the face plate of the lathe, but the outer 
surfaces and double spiral grooves were cut when the cylinder was mounted on a 
specially constructed mandrel supported between dead centres. The inner and outer 
surfaces of a cylinder are practically concentric and the ends at right angles to the 



axis. 



The turning of each cylinder occupied about four weeks. The cylinders referred 
to as Nos. 1 and 2 were turned by the late Mr. Taylerson of the Engineering 
Department of the Laboratory. Mr. Taylerson had marked ability for accurate 
work of this kind and made many useful suggestions during the progress of the work. 
The turning of cylinders Nos. 3 and 4 was very ably done by Mr. Tribe of the 
Engineering Department. 

The winding of the coils was carried out in a manner identical with that employed 
for the coils of the current balance. It is only necessary to state here that the coils 
were wound with the wire in tension, the effective load on the wire during winding 
being 4 kgr. The cylinders were rotated very slowly and after each revolution a 
stoppage was made for measurements to be taken of the diameter of the wire. 

The leads of all the coils must lie in a plane containing the axis of the cylinder, for 
otherwise the mutual inductance of the leads and the discs will not be zero. To 

ensure the absence of any correction due to such 

a cause, the connections to the coils were made in 

the manner shown in fig. 8. At the points where 

the coils terminate, two radial cylindrical holes 

a and b are drilled and enlarged to admit of the 

slotted brass nipples NN 7 being screwed into 

them. Soldered connections are made between 

the nipples NN', the leads c and d y and the leads 

through a and b. The leads c and d pass to two 

brass blocks mounted on an ebonite piece screwed 

to one end of the cylinder, and to these same 

blocks a small concentric cable is secured in the manner illustrated in the figure. The 

whole of the leads r shown in the figure lie in an axial plane of the coil and this same 

plane contains the leads connected to the other coil on the cylinder. 

An estimate was made of the accuracy with which the number of turns is known. 
On each cylinder there are two coils each having 96 turns and a diameter of about 
36 cm. ; the length of wire to each coil is, therefore, about 108 m. From observations 



N 









^v^ a K\x 



\\\\\\\n\\\\n 




\7 








Fig. 8. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 53 

on the radial holes and a consideration of the methods adopted for drilling them, the 
number of turns is considered to be correct within 2 parts in 1,000,000. 

The copper wire with which the eight coils are wound was supplied by The London 
Electric Wire Company on bobbins of the same diameter as the cylinders. It is hard 
drawn No. 24 S.W.G. and its mean diameter, obtained from about 800 measurements 
taken when winding the coils, is 0"557 5 mm. The diameter was also measured in the 
machine employed to determine the diameter of the coils, the mean of eighty measure- 
ments being identical with that already given. 

Measurement of the Mean Diameters of the Coils. 

Twelve axial planes at angular distances of 1 5 degrees apart are marked on the end 
faces and on the ungrooved portions of the outer cylindrical surfaces of each cylinder. 
These planes are numbered 1 to 12 and are the reference planes in the diametral 
measurements. Each cylinder was mounted in turn on the mandrel employed for 
turning the outer surfaces and supported between dead centres attached to the 
measuring machine. 

This machine was made by Sir W. G. Armstrong, Whitworth & Co. at their Openshaw 
Works, to designs prepared by the firm in collaboration with Mr. L. F. Richardson, 
formerly of the Metrology Division of the Laboratory. It consists of a straight bed 
carrying two fixed headstocks with cone-centres, between which the coil, on its 
mandrel, can be mounted. A saddle sliding on the bed carries two measuring head- 
stocks on a slide which is adjustable so as to bring the line of centres of these head- 
stocks exactly perpendicular to that of the fixed headstocks. Each of the measuring 
headstocks contains a barrel which can be moved in and out along the line of centres 
by means of a carefully calibrated micrometer screw ; and sliding freely in the centre 
of each barrel is a plunger, the front end of which constitutes the measuring face, 
while the rear end carries a small knife-edge pressing against a vertical lever 
pivoted in the barrel. At its upper end this lever carries a sensitive spirit 
level. The barrel is advanced by means of the micrometer screw until the 
plunger, being arrested by contact with the object to be measured, tilts the level 
far enough to bring the bubble to a definite mark. The reading is then taken 
with the aid of a vernier on the measuring wheel attached to the micrometer screw, 
to 100 * 000 inch. The same operation is carried out with each of the two measuring 
headstocks, one at either end of a diameter, and the sum of the readings is compared 
with the sum of similar readings on a gauge bar of known length. For convenience, 
this gauge bar is mounted on the machine during the measurements, so that by simply 
traversing the saddle from the coil to the gauge the zero reading can be checked at 
any stage of the work. To provide for this, one of the cone centres is made specially 
large, and pierced behind the cone by a hole through which the gauge can be passed. 

In the original design the control of the vertical lever carrying the level was effected 



54 



ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



by means of coiled springs contained in the barrel. It was found, however, that this 
arrangement gave rise to a greater pressure than was desirable at the point of contact 
between the plunger and the coil. Owing to the small radius of the wire with which 
the coils are wound, a comparatively slight pressure would produce deformation 
sufficient to lead to errors in the readings. A calculation showed that in order to 
keep errors due to this cause within allowable limits (less than x^o o mm - on "the 
diameter) the pressure must not exceed four or five ounces. At the suggestion of 
Mr. J. E. Sears, the springs were taken out and the vertical lever was extended 
downwards and provided with a weight at its lower end, thus forming it into a 
pendulum with a gravity control which could be adjusted to a nicety, At the same 
time the form of the body of the plunger was slightly modified, and its bearing 
surfaces lightly polished, so as to minimize friction. In this way the contact pressure 
was reduced to within the required limits. 

Many measurements of the coils were made with a current passing through them 
in order to reproduce as nearly as possible the conditions under which they would 
actually be used. To enable this to be done, the plungers were provided with non- 
conducting measuring faces of polished agate. These proved highly satisfactory in 
use, and the measurements made with the machine were of quite remarkable 
accuracy. 

Two pairs of Hartmann steel gauges were employed in the measurements. The 
following table gives their values at 0° C. : — 



Table II. — Giving the Values of the Gauges employed. 



# 



Approximate length of 
gauge in millimetres. 




By addition we get 



Gauges 300+ 60 
200 + 160 



Sevres value. 


- 


N.P.L. determination, 

using the Sevres value of 

the 1000 mm. gauge 

as basis. 


mm. fji 

60-1-5 

160-0*7 

200 + 0-1 

300-1-1 


mm. /x 
60-1-3 
160-0-7 
200 + 0*4 
300-0-9 


Sevres value. 
mm. /* 

360-2*6 

360-0-6 


N.P.L. value, 
mm. /x 

360-2-2 

360-0*3 



The Sfrvres values are stated to be accurate within 0'5/x, and the probable error of 



* fi = 1 micron = 0*001 mm. 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 



55 



the N.P.L. values is of the same order. By taking the means of Sevres and N.P.L. 
determinations as the best obtainable, we have 

300+. 60 == 360 mm.-2 , 4/x at 0° C., 
200 + 160 = 360 mm.-O'4/x at 0° C. 

Using the dilatation equation determined at Sfevres on a bar of the same material 

L, = Lo {1 + (10*469 + 0-00358*) t x 10~ 6 } 
we get the following lengths at 17° O. : — 

Gauges (300+ 60) = 360*062 mm., 
(200 + 160) = 360'064 mm., 

with an increase of 3"8/a on the 360 mm. length per 1° C. increase in temperature at 
or about 17° C. 

To obtain the mean temperature of the marble, holes were drilled in the cylinders 
and in these holes thermometers were inserted. The holes are parallel to the axis of 
the coils, and the depth of each is about half the axial length of a cylinder, so that 
the temperature recorded by a thermometer is very approximately the mean 
temperature of the cylinder. 

The temperature coefficient of expansion of the marble was determined by making 
diametral measurements at temperatures varying from 11° C. to 20° C. Assuming 
that the coefficient of expansion of the steel 
gauges is correct, the mean coefficient of linear 
expansion of the marble was found to be 
51 xl0~ 6 for the range 11° C. to 20° C. The 
observations were remarkably consistent, as will 
be seen from fig. 9, in which the difference of 
expansion of the steel and marble is plotted 
against temperature. During these observa- 
tions we were much struck with the compara- 
tive rapidity with which the temperature of the 
marble followed that of its surroundings. The 
temperature coefficient of expansion of the 
marble was also deduced from measurements 
of the axial length of a cylinder. In this case 
the difference of expansion of invar and marble was directly recorded, and taking the 
invar as known the value obtained for marble is 4*9 x 10~ 6 , which is in good agreement 
with the value given above. When required in our work we took the mean of 
these values, i.e., 5*0 x 10~ 6 , as correct. 

As already stated, the contact pieces of the measuring machine are of agate, in 



20jx 



5 

u 

fO 

z 

o 

z 



</) 

z 

i 

X 



fO 







I 

, , — i— _ .- — - —+/C — . - ,....._,, ■__ — ■ - ■. — — — 

_f 1 , ______ — „.., ., ..,. ■— — ■ — _ —■ ..■■->■■ 1 



I2L 



14 I© 18 

TEMPERATURE 



20 



22 



Fig. 9. Differential expansion, steel-marble. 



56 



ME. P. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



order that measurements of diameters may be made while a current is passing 
through the coils. The current used in our absolute measurements of a resistance 
did not exceed 2 amperes, and we experimented with such a current through the 
coils during our measurements of the diameters. Preliminary observations showed 
that the expansion of a cylinder 5 minutes after switching on the current was 
sufficiently great to allow of a definite measurement being made, and a thermometer 
recorded an increase of temperature of a whole degree in this time. A survey of the 
surface of the coil showed the expansion to be greatest in the mean diametral plane, 
and least at the ends ; measurements were, therefore, made on four turns of wire, 
these being the 1st, 16th, 48th, and 96th strands measured from one end. 



< 

u. 

o 

u 



< 

2 




10 



20 



30 



40 SO SO 

TIME IN MINUTES 



80 



Fig. 10. Showing increase in temperature of marble cylinder and increase in diameter of certain parts of 

the coils when a current of 2 amperes is passed through them for 37 minutes. 



indicates temperature observations 



x 



t 






observations on strand No. 96 

48 






J5 



JJ 



5J 



JJ 



r 



16 
1 



measured from one end of coil. 



The results of the measurements are given in figs. 10 and 11. In fig. 11 the 
expansion of the coil is given 5, 10, 15, 20, 25, and 35 minutes after a current of 
2 amperes was switched on. The maximum increase in diameter is 10*4^. Fig. 10 
shows the rise in temperature recorded by the thermometer when the current was 
left on for 37 minutes, and shows also the fall in temperature after the circuit was 
broken. A relation between the mean rise in temperature due to the current and 



EESISTANCE BY A METHOD BASED ON THAT OF LOBENZ. 



57 



the change in mutual inductance of the coils and the discs was worked out and found 
to be extremely useful when making the resistance measurements. The relation is 
as follows: — " If, in the resistance measurements, t x is the mean of the initial 
temperatures of the marble cylinders, and t 2 the mean of the final temperatures, the 
mean mutual inductance (during the resistance observations) of all the coils and the 
two discs is the same as when the temperature of the cylinders is uniform throughout 
and equal to ^ + 0*42 (* 3 — other things being kept constant." 




END 



CENTRE 



END 



Fig. 11. Showing expansion of coils on marble cylinders at intervals from 5 to 35 minutes, when a 

current of 2 amperes is passed through them. 



Complete measurements of the coils have been made on three occasions. The first 
set of measurements was made in February and March, 1912, and two sets were 
made in April, 1913. In the first set 192 observations were made on each coil, 16 
observations being made in each of 12 axial planes 15 degrees apart. The turns of 
wire measured in any one plane were 1 cm. apart, and neighbouring strands were 
measured in succeeding planes, so that one measurement was made on every turn 

of wire. 

In the measurements made in 1913, six measurements were made in each of 16 
diametral planes 1 cm. apart, alternate sets of six measurements being made in axial 
planes 30 degrees apart, and the remaining measurements were made in planes 
midway between these. In addition, 16 observations were made in each of three 
axial planes 60 degrees apart, thus making 144 observations in all. Both pairs of 
gauge bars were used, and the difference found between them is identical within 0'5/>c 
with that deduced from the values already given. 

In the following table we give the mean diameters of the coils on the four cylinders 
as deduced from the measurements made in February and March, 1912, and April, 
1913 :— 

VOL. COXIV. — -A. I 



58 



ME. E. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



Table III. — Giving the Mean Diameters of the Coils at 20°'0 C. 



Coil on cylinder 

No. 


February-March, 
1912. 


April, 1913. 


Mean. 


(a) 


0) 


1 

2 
3 
4 


cm. 

35-8808 3 
35-8817 6 
35-8855 4 
35-88674 


cm. 
35-8807 2 
35-8815 2 
35-8853 
35- 8866 2 


cm. 
35-8806 8 
35-8815 
35-8853 4 
35-88644 


cm. 
35-8807 4 
35-8815 
35-8853 9 
35-8866 



The values recorded in columns 2 and 3 are exactly the means of 192 and 102 
measurements respectively. The values in column 4 are not quite the means of the 
48 observations. The cylinders are not quite circular in cross-section, and the mean 
of 48 measurements in three planes should, in general, be different from the mean of 



Table IV. — Giving the Results of Measurements of the Diameters of the Coils on 

Cylinder No. 1. Temperature 20°'0 C. 



Axial plane No. . . 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 




Measurements on 
turn No. 


Diameter = 35 • 8750 cm. + 


Mean. 




fi. 


p. 


/*• 


/x. 


[i. 


/*• 


/x. 


/x. 


jjb. 


[A. 


p. 


p. 


fi. 


1 


29 




14 




22 




27 




30 




30 




25-3 


12 




27 




26 




31 




36 




36 




23 


29- 8 


24 


34 




30 




32 




36 




35 




35 




33> 


36 




35 




37 




42 




41 




47 




45 


41- 2 


48 


41 




39 




39 




46 




48 




49 




43 - 7 


60 




40 




42 




48 




47 




49 




48 


45- 7 


72 


61 




57 




59 




61 




64 




63 




60- 8 


84 




65 




66 




69 




70 




75 




65 


68- 3 


96 


67 




65 




76 




74 




72 




70 




70- 7 


108 




70 




69 




72 




69 




78 




66 


70- 7 


120 


58 




61 




63 




65 




72 




66 




64- 2 


132 




66 




65 




73 




74 




73 




68 


69-8 


144 


67 




63 




68 




69 




71 




72 




68- 3 


156 




66 




70 




74 




76 




70 




73 


71-5 


168 


75 




67 




73 




71 




72 




71 




71-5 


180 




65 




67 




70 




72 




65 




66 


67-5 


192 
Mean . „ . = 


64 




65 




70 




74 




78 




70 




70-2 


55-! 


53- 9 


51- 2 


55- 2 


55- 8 


59- 9 


58-x 


60- 6 


60-3 


61- 6 


58-4 


56- 8 


57- 2 



.\ Mean diameter (centre to centre of wires) = 35-8750 + 0*0057 2 = 35*8807 2 cm. at 20° -0 C. 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 



59 



measurements made in a large number of planes. This difference can be obtained 
from curves showing the cross-section of the cylinders, and in all cases we have 
estimated this difference and applied a correction. 

It will be observed that the means of the 1913 measurements are about 2/a less 
than those of 1912. This difference cannot be due to the gauge bars employed, and 
we believe it is not due to errors of observation ; it is probably due to a real 

Table V. — Giving the Results of the Individual Measurements on the same Coils, 

when Measurements were made in Three Planes only. 



Axial plane No. . . 


Between 3 and 4 


Between 7 and 8 


Between 11 and 12 




Measurements on 
strand No. 


Diameter = 35 • 8750 cm. 


+ 


Mean. 


■ Y 




/x. 


/x. 


fi. 


li. 


2 


24 


28 


31 


27'r 


12 


24 


29 


32 


28- 8 


24 


26 


32 


35 


31-o 


36 


32 


35 


38 


35-o 


48 


40 


42 


47 


43-o 


60 


42 


45 


47 


44- r 


72 


54 


66 


65 


61> 


84 


61 


66 


67 


64> 


96 


58 


61 


69 


62- r 


108 


67 


71 


74 


70> 


120 


57 


69 


63 


63-o 


132 


61 


67 


67 


65-o 


144 


56 


75 


70 


67-o 


156 


71 


79 


76 


75-3 


168 


67 


80 


74 


73> 


180 


68 


75 


71 


71*3 


192 
Mean . . . = 


70 


79 


75 


74- 7 


51*6 


58-5 


58- 9 


56-3 



.\ Mean diameter - 35 -8750 + 56*3/* = 35«8806 3 cm. at 20° *0 C. 

From fig. 12 and the observations in Table IV. we conclude that the mean diameter in these three 
planes is less than that of the complete coils by 0*5/x. Hence mean diameter of the coils = 35*8806 8 cm. 
at 20° -0 0. 



diminution in the diameters of the coils due to a displacement of small crystals of 
marble under the wires. This is quite possible, as the coils are wound under tension 
and have been subjected to considerable changes in temperature (due to the current 
passing through them) since 1912. 

Tables IV. and V. give the results of the individual measurements made in April, 
1913, of the coils on cylinder No. 1. 

I 2 



60 



ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



The curves given in fig. 12 show to what extent the coils vary in diameter. The 
conicality of the coils on cylinders 3 and 4 is much less marked than in the case of 
those on cylinders 1 and 2, but the variation from a circular cross-section is very 
small for all of the coils. The variation of diameter in any cross-section is so small 




ao 



T 



T 



j b i H i i > 



; > " ■ - ' - f 




it==it 




4 



5 6 



9 »0 n 12 i 



Fig. 12. 



AXIAL PLAN E N° 

Showing extent to which coils are conical, and variations in cross-section from circle. 



that the mean diameter of the cross- section may be employed for the purposes of the 
calculation of the mutual inductance without appreciable error, but the variation in 
diameter from end to end of the coils necessitates the application of a conical 

correction. 

The probable error of the mean diameter of the coils is estimated to be not greater 
than 1/x. The probable errors of the gauge bars used are stated on p. 54; the 
probable errors of the observations may be estimated from the data given in 
Table III. 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 61 

Measurement of the Axial Lengths of the Coils and the Variation of Pitch. 

The pitch measuring machine, also constructed by Sir W. G. Armstrong, Whitworth 
and Co. , consists of a straight horizontal bed provided with dead centres on which the 
mandrel carrying the coil can be mounted. Parallel to the line of centres is a slide, 
along which a small slide rest can be traversed by means of a calibrated measuring 
screw, of T Vinch pitch. The measuring wheel which actuates this screw can be read 
to 100 1 000 inch by means of a vernier, but the accuracy of repetition of readings is 
somewhat less than this. The slide carries a microscope whose optical axis intersects 
the line of centres of the machine at right angles. The microscope is focussed on the 
coil and is moved by means of the measuring screw so as to bring the consecutive 
turns of the coil, one after the other, into a definite position with respect to the cross 
wires in the eye-piece, the reading of the measuring wheel being taken at each setting. 
The calibration of the screw is effected by focussing a graduated line standard in the 
microscope and taking similar readings on the graduations of the bar. The measuring 
screw and slide are themselves movable on a second slide in a direction parallel to the 
axis of the microscope, so that by withdrawing them slightly the standard bar can be 
inserted at any time between the coil and the microscope, and check readings taken 
on it without disturbing the coil 

The standard employed was an invar metre, the history of which is so well known 
that the probable error over a length of 12 cm. (the axial length of a coil) is less 
than I//. 

For the measurements on each cylinder, readings on successive half centimetre 
divisions of the invar metre were taken over a length of 14 cm., and readings over a 
length of 12 cm. (corresponding to the axial length of the coils), were frequently 
made. Any change in temperature of the screw of the measuring machine could 
thus be allowed for. 

As already explained, the two wires on each cylinder are wound in double screw 
cut grooves. To measure the mean variation in pitch of the two coils it is better, 
therefore, to make observations on the spaces between neighbouring wires than on the 
wires themselves, for in the latter case twice as many observations are required. In 
practice, observations on the spaces proved the easier way ; the white marble showed 
up well between the wires, and it was not difficult to bisect the white spaces by means 
of the . cross wires of the microscope. In addition, check observations on the wires 
were always ,made. For a complete set of measurements 96 readings are required, 
and three such sets in different axial planes were made on each cylinder. 

After correcting the readings, the mean pitch of the coils was calculated within a 
few parts in 10,000 by dividing the difference of the extreme readings by the number 
of turns. The approximate pitch thus obtained was 0*065635 inch (= 0'166712 5 cm.). 
We multiplied this number by 1, 2, 3, 4, ..., &c, up to 95, and from the resulting 
products the corresponding pitch readings of a coil were subtracted. If the coil had 



62 



ME. F. E. SMITH ON THE ABSOLUTE MEASUEEMENTS OF A 



been perfectly uniform in pitch, the differences would, when plotted, have lain on a 
perfectly straight line, but, instead, wavy lines result (fig. 13). As an example of the 
difference readings those for the coils on cylinder No. 1 are given in Table VI. 

Table VI. — Giving the Difference Readings in Microns for the Coils on 

Cylinder No. 1. 



fl. 


fi. 


/x 8 


fi. 


16 


21 


36 


32 


11 


20 


31 


38 


7 


24 


36 


37 


12 


39 


47 


38 


28 


43 


51 


31 


31 


25 


42 


35 


23 


25 


36 


33 


17 


19 


25 


37 


5 


15 


27 


37 


8 


17 


35 


44 


3 


30 


45 


39 


23 


45 


53 


43 


27 


38 


40 


42 


37 


31 


35 


32 


27 


25 


29 


38 


22 


23 


23 


33 


10 


26 


28 


36 


12 


31 


33 


41 


15 


43 


45 


47 


28 


50 


44 


39 


46 


52 


41 


36 


42 


42 


34 


44 


35 


40 


30 


43 


28 


31 


27 


37 



From the products resulting when the numbers 1, 4, 7, 10, &c, up to 94 were 
multiplied by 0*065635, the half centimetre readings for the invar metre were 
subtracted and the differences when plotted were found to lie on a straight line as 
they should do if all the corrections have been properly applied. The graphs for the 
invar metre are shown in fig. 13 and are distinguished as Invar 1? Invar 2 , &c, the 
numbers corresponding to the numbering of the four cylinders. A phosphor-bronze 
ring is screwed into one end of each cylinder, and these ends are called the " ring ends." 

It is clear from fig. 13 that there is a periodic variation in the pitch of all the coils. 
The distance corresponding to a complete cycle is half an inch, which is equal to the 
pitch of the leading screw of the lathe on which the cylinders were turned. The 
cause of the periodicity may be in the leading screw itself, or it may be due to want 
of parallelism of the surfaces at which the thrust was taken, or it may be due to both 
of these. Such periodic variations in pitch are present in nearly all screws. In 
addition to these periodic irregularities, the graphs show others which are taken into 
consideration when the mutual inductance is calculated of a coil and disc 
circumference. 

The graphs for the coils enable one to place the cylinders in pairs. The lengths at 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 



63 




w 

I— I 

• r-l 
O 

o 

O 
O 

• r-l 



e! 
o 

• r-l 

c8 

*> 

a 
•in 

o 

QQ 



CO 

I-H 

&b 

• r-l 



HOJLId NV3N JO NOIllSOd 

tNOdJ 3fclM JO XN3lN30VldSIQ 



64 



ME. F. E. SMITH ON THE ABSOLUTE MEASUEEMENTS OF A 



20° '0 C. of the coils on cylinders Nos. 1 and 2 are 16 '00 18 cm. and 16'0013 5 cm. 
respectively; those on cylinders Nos. 3 and 4 are 16*0054 cm. and 16'0054 cm. 
respectively. The amplitude of the periodic variation dies away towards one end for 
the coils on cylinders Nos. 1 and 2, but for the coils on cylinders Nos. 3 and 4 it does 
not. The graphs for the variations in diameter of the coils lend support to this 
division of the cylinders into pairs, and as the same lathe and the same portion of the 
leading screw were used for the turning of all, the differences appear to be due to a 
slight difference in the skill and touch of the late Mr. Taylerson, who turned 
cylinders Nos. 1 and 2, and Mr. Tribe, who was responsible for cylinders Nos. 3 and 4. 
In all cases the turning is remarkably good. 

The mean axial length of the coils on one cylinder is obtained from the graph 
showing the periodic variation of the pitch and the corresponding graph for the invar 
metre. Limiting our attention to the coils on cylinder No. 1, it is clear that the 
extreme mean length of the coils is equal to the length of the invar metre section 
AB plus the length corresponding to the difference of the ordinates OA and PB. 
The length of the invar section is 15'9996 2 cm. and the difference of the ordinates 
corresponds to +17/*. Hence the mean length of the coils on cylinder No. 1 is 
15'9996 2 +17m = 16'0013 2 cm. at 14 0, 5 C. = 16*0018 cm. at 20°'0 C. 

Table VII. gives the mean axial length of the coils at 20° '0 C. 

Table VII. 



Coils on cylinder. 


Mean axial length at 20° • 0. 


1 
2 
3 
4 


em. 
16-0018 
16-0014 
16-0054 
16-0054 



These observations for the mean axial length were made in April, 1913. The 
results are identical within the limits of the errors of observation (about 3/x) with 
those obtained in March, 1912. A change in the length of a coil of 20^ produces a 
change in the mutual inductance of that coil and the two discs of 1 part m 100,000. 

Section 14.— Erecting and Adjusting the Instrument. 

Before assembling the parts of the machine, a concrete block, built up of Keene's 
cement and Thames ballast, was prepared as a foundation. The block is non-magnetic, 
is 8 m. long, 80 cm. wide, and 120 cm. deep. Slide rails of gunmetal are bolted in 
position on this block and to secure greater stability the rails are sunk 5 cm. into the 
concrete. The rails are in three pairs ; one pair support the motor ; a second pair the 
fly-wheel ; and the third pair, which are nearly 4 m. long, support the portions with 
the rotating discs. A period of twelve months was allowed for the concrete block to 



EESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 65 

assume approximately constant dimensions, and at the end of that time the upper 
surfaces of the rails were scraped to obtain flat surfaces and to ensure these surfaces 
lying in the same horizontal plane. Two special fitters from Sir W. G. Armstrong, 
Whitworth & Co. carried out this part of the work, and also superintended t|he 
alignment of the parts of the machine. The under surfaces of the castings 
supporting the fly-wheel, discs, &c, were scraped plane at the Elswick Works and the 
alignment of the parts was, in consequence, a comparatively easy task. Originally, 
the couplings were intended to be flexible ones, but after a few runs of the machine 
rigid couplings were found to be better and such were used. Much of the work 
connected with the couplings and other fittings to the machine was carried out in the 
Engineering Department of the Laboratory, and we are greatly indebted to 
Dr. Stanton, the Superintendent, for much advice on these matters. 

At first there was trouble with the bearings. The clearance allowed was very small, 
and after the machine had run for one or two hours the expansion of the shaft was 
sufficiently great to cause a collar to come into contact with one of the bearings. 
Increased clearance was allowed and the difficulty disappeared. From that time 
(April, 1911) no portion of the machine has given the slightest trouble. The parts 
are so exactly balanced that the tremor of the concrete base is almost too small to be 
detected. This is fortunate, as during the observations for an absolute measurement 
of a resistance, two microscopes are mounted on the concrete bed and used to gauge 
the distance apart of two coils within a thousandth of a millimetre. These measurements 
were frequently made when the speed of the machine was 1040 revolutions per 
minute, but an accuracy within a thousandth of a millimetre was obtained with little 
trouble. During the progress of our work the machine has been admired by many 
hundreds of visitors to the Laboratory, and the kindness of Sir Andrew Noble in 
having the heavy metal work carried out at Elswick is greatly appreciated. 

To place the cylinders in position on the cradles, the portions of the shaft supporting 

the discs were removed from their bearings, the cylinders threaded over the shafts 

into approximately correct positions and the shafts restored to their proper places. To 

prevent damage to the cylinders during this operation, a framework of wood was built 

to support the shafts and cylinders and to enable them to be lifted as one piece. The 

cradles on which the cylinders are supported are provided with stops and clamping 

screws to prevent any marked relative movement of a cylinder and the cradle on which 

it rests. 

Section 15. — Insulation Tests. 

When making an absolute measurement of resistance, the greatest difference of 
potential between any portion of the circuit and the earth was about 130 volts, and 
the greatest difference of potential between neighbouring turns of wire on any one 
cylinder was about 16 volts. 

The insulation resistance between the circuit and the earth was tested each day on 
which measurements of resistance were made. For the insulation test, the earth wire 

VOL. CCXIV. A. K 



66 MB. F. E. SMITH ON THE ABSOLUTE MEAStJEEMENTS OF A 

(p. 85, fig. 19) was removed, and the voltage applied to the circuit was the same 
as that used for the resistance measurements. The insulation resistance was always 
greater than 1000 megohms. 

To measure the insulation resistance between the coils, the latter were usually arranged 
in two groups, each group consisting of one coil from each cylinder. At first, with an 
applied voltage of 20 volts, the insulation resistance was about 10,000 ohms. The 
insulation resistance of coils on cylinders Nos. 2, 3, and 4 proved to be well above 
1000 megohms, but that on the coils on cylinder 1 was low. We found these coils to 
be in close proximity to, or in contact with, a few crystals of pyrites, and after 
dislodging these, wholly or in part, the insulation resistance increased to above 1000 
megohms. No trouble has since occurred. As the coils are of bare copper wire and 
are covered only with a thin wrapper of silk, we think it necessary to make an 
insulation test of the coils on each day that resistance measurements are made. The 
test occupies but a few minutes, and during our work it was regularly made. 

The insulation resistance between the earth and the rotating wires attached to the 
discs was usually tested at 20 volts. The results in all cases were satisfactory. For 
certain measurements, it is essential that the rotating wires be insulated from each 
other and as the wires may not occupy exactly the same positions with respect to the 
disc and shaft when rotating, as when stationary, the tests are preferably made with 
the machine running. The test when the wires are rotating is of some interest. One 
terminal of the galvanometer is connected to one brush, and one pole of the battery to 
the remaining four brushes, the five brushes being in contact with the segments on 
one of the discs. The other terminal of the galvanometer is connected to the remaining 
pole of the battery. From fig. 5, given on p. 42, it is clear that when the discs are 
rotating one terminal of the galvanometer is always connected to one or two of the 
rotating wires, and the other terminal, through the battery, is connected sometimes 
with four and sometimes with eight of the wires. The wires are continually changing 
in position and the deflection of the galvanometer enables the insulation resistance to 
be calculated. The first set of conducting wires which were used became faulty in 
insulation resistance because of the action of some insulating tape which was used to 
bind the wires together. They were replaced by others which were supplied with 
double layers of silk and which we subsequently shellacked and encased in silk tubes. 
No trouble has since been experienced, the insulation resistance being well over 200 
megohms. The insulation of the remaining part of the circuit, e.g., leads to brushes, 
standard resistance, &c, was frequently tested, but no fault was found. 

Section 1 6 .— Me asurement of the Diametral Distance between 

Opposite Segments. 

The distance between opposite segments was measured both when the discs were 
stationary and when running at speeds varying from 170 to 1110 revolutions per 
minute. 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 



67 



For the stationary measurements, two micrometer heads were fixed to, but 
insulated from, two short upright rods of brass secured to a stout bar of the same 
metal, the distance between the contact faces of the micrometer screws being 
approximately equal to the distance between opposite segments. This gauge was 
supported on two uprights secured to the slide rails of the Lorenz apparatus and 
adjusted in position for the measurement of a diameter. Between the segments and 
the micrometer screws, vertical wires of phosphor bronze, similar to those used for the 
brushes, were interposed, and the measured distance was taken as equal to the 
distance apart of the segments plus twice the diameter of a wire. Contact between 
a segment and a micrometer head was indicated by the buzzing of a telephone due to 
the passage of a small alternating current through a circuit including the telephone, 
the micrometer head, and the segments. Four measurements were made on each pair 
of segments lying on opposite sides of a diameter and the mean of the 20 measure- 
ments was taken as correct. Such a series of measurements was frequently taken 
and show the wear of the segments, due to their friction with the brushes, to be 
comparatively slight. As examples, we give the results of some measurements made 
in December, 1912, and in June, 1913. 

Table VIII. — Giving the Diametral Distance between Opposite Segments. 



Segments. 


Disc No. 1. 


Disc No. 2. 


December, 1912. 


June, 1913. 


December, 1912. 


June, 1913. 


1 and 6 

2 „ 7 

3 „ 8 

4 „ 9 

5 „ 10 


cm. 
53-581 
•579 
•579 
•579 
•584 


cm. 
53-562 
•566 
•569 
•573 
•570 


cm. 
53-556 
•556 
•550 
•556 
•554 


cm. 
53-546 
•547 
•546 
•549 
•560 


Mean radial distance ~) 
of segments . . J 




53-580 

v . 


53-568 


53-554 

i 


53-550 


53-574 cm. 
= 26-787 cm. 


Y 

53-552 cm. 
-26*776 cm. 



The standard of length used was an invar rod with approximately flat ends, and 
measurements on this rod were made in the same way as those on the segments. 

When the discs were rotating, contact with the segments was made by advancing 
a brush by means of a micrometer screw until a telephone indicated the completion of 
a circuit as in the previous measurements. The micrometer head made contact with 
the brush holder through the medium of a small steel hemisphere attached to the 

K 2 



68 



MR F. E. SMITH ON THE ABSOLUTE MEASUEEMENTS OF A 



holder, and good contact was ensured by a spring attached to the supporting ring and 
brush, which forced the latter towards the micrometer screw. The brush wires were 
put under considerable tension and petrol was used as a lubricant. Measurements 
were made at speeds varying from 170 revolutions per minute to 1110 revolutions per 
minute, the speed being registered by the directly driven chronograph. The results 
of three sets of measurements are plotted in fig. 14. The relation between the 







































MM. 

0*05 






































































if) '04 

D 

Q 


























o 


■ 


























" © 


« 














< 


















/ '/ 


















03 

Z 
















r* 


r t 


















£ 02 
< 












































% 


1____4 


5 
























Z 'Ol 










o 




























ifc/6 


■"o 



































































o 



a 

N 1 K 



IO 



«Z 



10" 



N= N° 



>F REVOLUTIONS 

Fig. 14. 



14- 



PER MIN 



increase in radius and the square of the number of revolutions per minute is 
dr = 3'7n 2 x 10~ 8 , where dr is the increase in radius in millimetres and n is the 
number of revolutions per minute. The normal speed of the apparatus when resist- 
ance measurements are in progress is about 1050 per minute, and the mean radius is 
then 0*04 mm. greater than the mean value given in Table VIII. The effective 
radial distance at 20°'0 C. of the segments is therefore 26787 + 0*004 = 26791 cm. 
for disc No. 1 and 26776 + 0'004 = 26780 cm. for disc No. 2. 



Section 17. — Determination of the Distance between the Centres 

of the Coils. 

The distance between two coils on opposite sides of a disc has to be known with 
considerable precision, as a change of three-thousandths of a millimetre in this distance 
changes the mutual inductance of the coils and discs by 1 part in 100,000. As it is 
not possible to make direct measurements of this distance with both rapidity and 
precision, we followed the plan adopted by Glazebrook # (and by Lord Rayleigh in 
1882) of reversing the coils, without interchange, and then repeating the resistance 



* i 



Phil. Trans.,' vol. 174, p. 251, 1883. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 



69 



measurements. Reference marks are made on the flanges of the cylinders , and when 
the coils are parallel the mean distance between two such marks, one on each cylinder, 
in the two positions of the coils, is. exactly equal to the mean distance between the 
mean planes of the coils. If there are two marks diametrally opposite on each cylinder 
and two distances are measured for each position of the coils, the mean distance between 
the marks is the mean distance between the centres of the coils if the latter are 
approximately parallel. In our case a want of parallelism of 0*25 degree (far in excess of 
that met with in practice) introduces an error of less than 1/x, in the determination of 
the distance between the centres of the coils. In practice we have four marks on 
each cylinder, these being at approximately equal distances from the mean plane of 
a coil, and the distance of each mark from such plane is known within 10/x, or 20/x. 
The vertical distances of the marks from the axis 
of the coil are the same within O'l mm., and for 
the measurements under consideration the dis- 
tances may be regarded as identical. 

Fig. 15 represents diagrammatically two cylin- 
ders at a mean distance 00 ; apart. When the 
coils are parallel, the distances ac, eg, bd, and fh 
are approximately equal to 00 ; . When the coils 
are reversed but not interchanged, the distances 
are in general different from what they were before ; 
let the distances be aV, e f g\ &c. With parallel 
coils each of the mean distances (ac + aV)/2, (eg + e'g')/2, &c, is equal to the average 
distance apart of the mean planes of the coils. If the two coils are not parallel but 
very nearly so, the average of the distances ac, a!c\ eg and e f g r is equal to the average 
of the remaining four distances and also equal to the average distance apart of the 
centres of the coils. 

In the flanges of each cylinder four brass plugs are screwed and cemented in 
position. The plugs in each flange are at opposite ends of a diameter, and the four 
plugs in one cylinder are contained in a common axial plane. Fig. 4 shows two of the 
plugs ab in cylinder No. 3, and two cd in cylinder No. 4. On the faces of the plugs 
thin rectangular pieces of platinum are soldered ; the surfaces of these are polished, 
and a scale of half millimetres, cut at right angles by two horizontal lines, half a 
millimetre apart, is engraved on each piece. The scales were engraved by the late 
Mr. Donaldson of the Metrology Department. 

For the measurement of the distances, two microscopes are mounted on a special 
platform supported on a tripod, the base of which can be clamped to the bed of the 
Lorenz machine by means of the wing nuts w 9 w\ w'\ &c, shown in fig. 4, and by 
means of screws the platform can be given the necessary movements to enable both 
microscopes to be rapidly focussed without any change in their relative position. 
When the platform is clamped by the wing nuts 'ww\ the lines on the plugs a and c 




70 



ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



are under observation, and when, the platform is clamped by means of w f and w", the 
lines on plugs b and d can be viewed, and so on for the remaining positions. 

The reference standard is a scale of half millimetres, engraved on invar. The scale 
was standardised by Mr. Attwell before making any absolute measurements of 
resistance, and again after their completion. The results on the divisions employed 
in our work agree within less than 1/x. 

The general procedure in making the measurements was to observe the invar scale, 
then the eight plug distances, and finally the invar scale again. The first and last of 
the observations agreed in general within 1/m or 2/x. An absolute measurement of 
resistance immediately followed and then another set of observations was made on 
the invar scale and the plugs. The coils were not reversed in position until a large 
number of resistance measurements had been made. 

The distance between coils Nos. 1 and 4 and between Nos. 2 and 3 is required with a 
much less degree of accuracy, an error of 1 mm. in both of these distances producing a 
change in the mutual inductance of less than 1*5 parts in 100,000. As the distances 
between coils Nos. 1 and 2 and Nos. 3 and 4 are known from the previous measurements 
within a few thousandths of a millimetre, it was only necessary to measure the 
distance 1 to 4, or 2 to 3. In practice we measured the distance between a plug on 
cylinder 2 and a plug on cylinder 3, and employed for this purpose a brass bar with 
half-millimetre divisions engraved at the ends ; this enabled the distance to be 
determined within about 0M mm. The distances of the plugs from the mean diametral 
planes of the coils were known and no further measurements were therefore necessary. 
Instead of measuring the distance between two coils such as Nos. 2 and 3, we may, as 
an alternative, measure the distance between the two sets of five brushes, and a number 
of such measurements were made as checks. 



Section 18.— Calculation of Mutual Inductance of the Coils and the 

Contact Circles of Segments and Brushes. 

The arrangement of the coils and discs in the instrument may be represented 
diagrammatically in section by fig. 16. The coils are numbered 1, 2, 3, and 4, and the 
discs are denoted by D 1 and D 2 . 

K 



A 



B 



C E 



H 




D. 



D 



Fig. 16. 



The mutual inductance M of any one coil and the contact circles is calculated in two 
parts. The first part gives the mutual inductance M x of the coil and the circumference 



KESISTANCE BY A METHOD BASED ON THAT OF LORENZ. fl 

of the disc nearer to it, and the second part gives the mutual inductance M 2 of the 
coil and the circumference of the other disc. The difference of these is required, so 
that M = Mi-Ma. 

To find M 1? two mutual inductances were calculated, viz., that between D x and a 
coil of length BC, and that between D 1 and a coil of length AC. The difference of 
these mutual inductances gives M x . If the coils on the four cylinders are exactly- 
similar the values of M x are identical. 

To find M 2 for coils Nos. 1 and 4, the mutual inductances were calculated between 
D 2 and a coil of length BK and that between D 2 and a coil of length AK. The 
difference is equal to M 2 . To find M 2 for coils Nos. 2 and 3, the mutual inductances 
required are that between D 2 and a coil of length EK and that between D 2 and a coil 
of length FK. 

In the Lorenz apparatus, the distances of the coils from the discs can be varied. 
This changes the mutual inductance and the rate of change of M with variation of 
axial distance must therefore be known. 

To find M x we used the following formula due to J. Vireamu Jones # 

M = Q(A + a)ck l^~ + ^(F-II) 

In this expression, is the angular length of the helix, A the radius of the helix, 
a the radius of disc or contact circle, and x the axial length of helix. 

c 2 = 4Aa/(A + a) 2 , c' 2 =l-c 2 , 
W = 4 Aa/( A + of + x\ ¥ 2 = 1 - k\ 

F, E, and II, are complete elliptic integrals of the first, second, and third kinds 
respectively ; F and E are to modulus Jc, and 

n = p ^ 

Jo (l—c 2 sr 



(l-^sin^Hl-Fsin 2 ^) 1 ' 



Putting c f jM = sin /3, the quantity (F— II) can be expressed in terms of complete 
and incomplete integrals of the first and second kindsf ; thus 

c- 1 ^ 2 sin £ cos P (F-II) = - ^tt-F (*) F (&', $) +E (jfe) F (k\ 0) +F (*) E (&', /J). 

The various elliptic integrals required were calculated by interpolation from 
Legendre's tables, but as a check on possible misprints in the tables a number of the 
integrals were calculated directly by successive quadric transformation.! 

The dimensions chosen for calculating the values of M x are given in Table IX. The 

* J. V. Jones, 'Koy. Soc. Proc./ vol. 63, p. 198, 1898. 
t Cayley, * Elliptic Functions/ § 183. 
t Cayley, Chapter XIII. 



72 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



dimensions of the coils and discs differ from these values by small amounts dA for 
the radius of the coils, da for the radius of the discs, and dx for the length of the 
coils. Small corrections to the calculated values in Table IX. have therefore to be 
applied. These corrections are obtained by application of the increment formula 

dM dA , da , dx 
M * A* a ■ x 

which gives the change in M due to small changes in dimensions, A being the radius 
of the coil, a that of the disc, and x the axial length of the coil ; g, r, and s are 
coefficients which are given by the expressions 



Qck 






g=_A|F+^(F-II)J 



M 



\jCfv TQ1 XjL a /T71 TT\ 



M 2A 



«l.®^(A + a )jfl-^F-?,E 



M 



¥ 



¥ 



The sum q + r + s is always equal to unity. 

In Table IX. we give in columns 1, 2, and 3 the constants employed for the 
calculation of nine mutual inductances ; in column 4 the values of the mutual 
inductances ; and in columns 5, 6, and 7 the values of g, r, and s, which are required 
in our work. Table X. gives the difference values M x . 



Table IX.— Calculation of Mutual Inductance. 

A = radius of coil, a = radius of circle (disc), x = length of coil. 

The number of turns per centimetre length of the coil is 12. 



A. 


a. 


*v. 


M. 


1 


f. 


5. 


cm, 


cm. 


cm. 


cm. 








-■■■. 


17- 


9419 


26' 


7870 


23-2650 


52702' 


37 


2*1003 


-0-5737 


-0-5265 


17- 


9419 


26' 


7870 


7' 


2635 


23682- 


39 


2-4007 


-1-2626 


-0-1381 


17' 


9419 


26' 


7870 


23' 


3150 


52755 


•93 


2-0998 


-0-5724 


-0-5274 


17' 


9419 


26 


•7870 


7' 


3135 


23822 


•74 


2-3991 


-1-2596 


-0-1396 


17 


•9419 


26 


•7870 


23' 


2900 


52729 


•18 


2-1000 


-0*5731 


-0-5270 


17 


•9419 


26 


•7870 


7' 


2885 


23752 


•65 


2-3999 


-1-2611 


-0-1388. 


17 


•9419 


26 


•7870 


19 


•285U 


47883 


•64 


2-1404 






17 


•9419 


26 


•7870 


11 


•1678 8 


33534 


•20 


2*2880 






17-9419 


26-7870 


15-2572 5 


41656-11 


2-2001 




i 



In Table X. the first difference values are the mutual inductances of a coil 
16*0015 cm. long and a circle of radius 26*7870 cm., when the mean diametral plane 



EESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 



73 



of the coil is distant from the circle 15'2642 5 cm., 15'2892 5 cm., and 15*3142 5 cm. 
respectively. In these three positions an increase in the radius of the coil of 10 jut, 
increases the mutual inductance by 3'001, 2*995, and 2*988 respectively, and an 
increase in the radius of the circle of 100 /x increases the mutual inductance by —0*125, 
— 0*099, and —0*072 respectively. If we have two such coils, one on each side of the 
circle, the intensity of the magnetic field at the circumference of the circle will be 
zero when the mean planes of the coils are about 30*656 cm. apart and the circle is 

Table X. — Difference Values of Mutual Inductance. 



A = radius of coil 



a = radius of circle . 



x = Axial length of coil . 



M = mutual inductance 
Difference value = Mi . 



dM. for elk = 10/x 
Difference value . 



dM. for d A - 100/x . 
Difference value . . 



17-9419 



26-7870 



23-2650 



52702-37 



17-9419 



26-7870 



7-2635 



23682-39 



29019-98 



+ 6-169 



+ 3-168 



+ 3-001 



11-288 



11-163 



Y~ 



0-125 



17-9419 



26-7870 



23-2900 



52729-18 

v. 



17-9419 



26-7870 



7-2885 



23752-65 



28976-53 



+ 6-172 



+ 3-177 



— y- 



_J 



+ 2-995 



-11-281 



— V" 



-11-182 

> 



-0-099 



17-9419 



26-7870 



23-3150 



52755-93 



17-9419 



26-7870 



7-3135 



23822-74 



28933-19 



+ 6-174 



+ 3-186 



^r 



+ 2-988 



-11-274 



-11-202 



0-072 



midway between them. The same result is of course brought about by a slight 
reduction in the diameter of the brush contact circle, an inevitable result of wear and 
re-turning of the surface of the segments. However, exact realisation of this condition 
is unimportant since for the positions of the coils dealt with in the calculations, the 
maximum change in the mutual inductance is only 4 parts in 1,000,000 for a change 
in the diameter of the circle of one-fifth of a millimetre. Exact centering of the 
circle between the coils is also unimportant as will now be shown. 

When the distance of the circle from each of the mean diametral planes of the coils 
is exactly 15*2892 5 cm. the total mutual inductance of the two coils and the circle is 
57953'0 7 . If the circle is moved so as to be 0*25 mm. nearer one of the coils the 
total mutual inductance is increased to 57953*1 7 , i.e., about 1 part in 600,000 greater 
than before. Hence with a brush half a millimetre wide the variation of potential 
from wire to wire will not exceed 1 part in 600,000. In practice the brushes 
commonly used consisted of three parallel wires, the extreme width being about 
0*5 mm. 

vol. ccxiv. — A. 



L 



74 



ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



Table XL gives the constants used and the results obtained in one of the calculations 
dealt with in Table X. and gives also the dimensions of the Lorenz coils and brush 
contact circles at 20°"0 C. Table XII. gives (l) the differences of dimensions of the 
coils and standard, and (2) the mutual inductance M x of each coil and the contact 
circle nearer to it. 





Standard. 


Coils on cylinder No. — 


1. 


2. 


3. 


■ 4. 


A = radius of coil in centimetres . 

x = length of coil in centimetres . 

a = radius of circle in centimetres 

d = distance of mean diametral! 
plane of coil from the circle > 
in centimetres .... J 

n = number of turns ..... 

Mi = mutual inductance of coil") 

and circle ...... J 


17-9419 
16-0015 

26*7870 

15-2892 5 

16*0015x12 
28976*54 


17-9403 r 

16-0018 

26*791 

192 


17*9408 

16-0014 

26-791 

192 


17-9427 

16*0054 

26*780 

192 


17*9433 
16*0054 

26*780 

192 



We summarise below our knowledge of the changes of M x with changes in the 
dimensions of the coils and disc when the distance of the mean diametral plane of the 
coil from the brush contact circle is equal to 15*2892 5 cm. 



1. Increase in radius of a coil by 10/x . . . . . . 

2. Increase in radius of contact circle by 10/x . . . 

3. Increase in length of coil by 10/*, the position of the 

mean diametral plane remaining constant, and the 
number of turns increased so that there may still 
be 12 turns per centimetre . . . 



Change in M x . 
+ 2*995 

-0*0099 



~\ 



5, 



+ 1-940 



J 



4. Correcting factor to reduce the mutual inductance \ 16/length of coil in 
to a coil of exactly 192 turns is ...... J centimetres. 



The values of M l given in Table XII. hold good only when the mean diametral plane 
of the coil is 15*2892 5 cm. from the circle. For other slightly different distances 
we may readily calculate the corrections from the data given in Tables IX. and X. 
The mutual inductance of the coils on cylinders Nos. 1 and 2 and the brush contact 
circle No. 1 is 579397 when the distance between the mean planes of the coils is 
30*5785 cm., and the corresponding values for the coils on cylinders Nos. 3 and 4 is 



KESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 



75 



57955*4, the temperature being 20°*0 C. For an increase in the distance apart of the 
mean planes of Nos. 1 and 2 by 250^ the mutual inductance diminishes by 43*43, and 
for a decrease in the distance by the same amount the mutual inductance increases 
by 43*32. The corresponding values for cylinders Nos. 3 and 4 are practically 
identical, being 43*48 and 43*37 respectively. 



Table XII. — Giving the Mutual Inductance M a of the Coils and Contact Circles on 
the Assumption that the Hadius is at all Points Equal to the Mean Radius and 
that the Pitch is Uniform. 



Coils on cylinder No. — 


1. 


2. 


3. 


4. 


Differences. 


Correc- 
tion 
to Mi. 


Differences. 


Correc- 
tion 
to Mi. 


Differences. 


Correc- 
tion 
to Mi. 


Differences. 


Correc- 
tion 
to Mi. 


fjb. 
dk. =--15-3 

dx = + 3 
da = +40 

For number V 
of turns n J 

Total 1 
correction J 


-4-58 

+ 0-58 

. -0-04 

-3'26 
-7-30 


dk = -11 
dx = - 1 
da = +40 


-3-29 
-0-19 
-0-04 

-2-54 
-6-06 


dk = + 8 
dx = +39 
da = - 70 


+ 2-40 
+ 7-57 
+ 0-07 

-9-78 
+ 0-26 


ft. 

dk = +14 
dx = +39 
da = - 70 


+ 4-19 

+ 7-57 
+ 0*07 

-9*78 
+ 2-05 


Mi = 28969-24 


Mi = 28970-48 


Mi = 28976-80 

V . ,,. ,. . ,-,. 


Mi = 28978-59 


Sum -57939*7 


Sum - 57955-4 



With change of temperature the mutual inductance varies owing to the expansion 
of the coils and discs. If the distance apart of the mean diametral planes of the coils 
is kept constant, the temperature coefficient of M 1 is readily calculated to be 1*06 x 10~ 5 
for an increase in temperature of 1° C. 

We have previously stated that it is not necessary to know the axial length of a 
coil with great accuracy. From the data given on page 74, it is clear that if the 
coils increase in length by 10/*, the change in M x for the coils on any one cylinder is 
+1*94. But this supposes the number of turns to increase in the ratio 16*0010/16*0000. 
Since the number of turns is constant the real change is 1*94 + 28970— 28970 x 
(16*0010/16*0000) = 0*13. That is, a change of 10 M in the axial length of the coils 
produces a change in the mutual inductance of the coils and discs of about 5 parts in 
1,000,000. 

L 2 



76 ME, F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

Calculation of M 2 , the Mutual Inductance of the Coils > and the farther Brush 

Contact Circle. 

The following formula, due to Rosa and Grover,^ gives the mutual inductance 
of a single layer coil of length x and a co-axial circle of radius A in the plane of one 
end of the coil : — 



M = 2A 2 N 



3a 2 A 2 5a 4 AV 35q 6 A 6 y 63a 8 A 8 
8 d" 64 d 8 2 512 d 12 4 1024 cP 



d 
where 

X 2 - 3-4^/A 2 , 

X 4 = 5/2-1 (te 2 /A 2 + 4^ 4 /A 4 , 

X 6 = 35/l6-~35^ 2 /2A 2 +21^ 4 /A 4 -4^7A 6 ? 

N = total number of turns in length x, 

a — radius of coil, 
A = radius of circle, 

x = axial length of coil, 

d = vV + A 2 . 

When the coil is long this formula is very exact and easy to use, and it was a 
simple matter to calculate M 2 with the precision necessary for our work. The results 
of 16 calculations are given in Table XIII. 

In Table XIV. the values of M 2 are given for coils, the mean diametral planes 
of which are distant from the brush contact circle by the amounts given in column 4. 
The results show the variation of M 2 with change (l) in the axial distance of the 
coils, (2) in the radius of the coils, and (3) in the radius of the contact circle of 
segments and discs. 

The summary on p. 78, Table No. XV,, relates to the coils and brush contact 
circles of the Lorenz apparatus and sufficiently indicates how small corrections were 
made when the distance between the coils was varied. The contact circle of segments 
and brushes was practically in the midplane of the coils near to it ; the method of 
ensuring this is described in Section 22. 

Correction for Conicality of Coils. 

The increment coefficient q (Table IX.) enables the change of mutual inductance to 
be calculated when the radius of a coil is changed by a small amount, but the change 
must be a uniform one. When the change is not uniform, the change in mutual 
inductance for an increase in radius of any part of a coil must be known, and the 
correction for conicality must be calculated in parts. In the past it has been 
customary to take the radius of a coil as absolutely uniform from end to end, but 
such a procedure invariably introduces errors into the calculation. 



* i 



Bur. of Standards Bull.,' vol. 8, No. 1, p. 101. 



RESISTANCE BY A METHOD BASED ON THAT OF LQRENZ. 



77 



Table XIII. — Calculations of Mutual Inductance. 



Radius of 
circle. 



cm. 

26-7870 

26-7870 
26-7870 
26-7870 
26-7870 
26-7870 
26-7870 
26-7870 
26-7870 
26-7870 
26-7870 
26*7870 
26-7370 
26-7370 
26-7370 
26-7370 



Radius of coil 

of 12 turns 

per centimetre 

length. 



cm. 
17-9419 

17-9419 

17-9419 

17-9419 

17-9419 

17-9419 

17-9419 

17-9419 

17-9369 

17-9369 

17-9369 

17-9369 

17-9419 

17-9419 

17-9419 

17-9419 



Length of 
coil. 



cm. 
144-1 

160-1 

174-6 

190-6 

144-4 

160-4 

174-9 

190-9 

144-1 

160-1 

174-6 

190-6 

144-1 

160-1 

174-6 

190-6 



Mutual 

inductance of coil 

and circle. 



74981-04 
75215-30 
75376-21 
75514-07 
74986-12 
75219-06 
75379-13 
75516-33 
74939-24 
75173-38 
75334-19 
75471-98 
74985-66 
75219-08 
75379-42 
75516-78 



Mutual 
inductance 
differences 

= M 2 . 



234-26 



r 



137-86 



232 


•94^ 




> 


137 


•20 



234-14 



137-79 



233-42 



137-36 



Sums of 
differences. 



372-12 



370-14 



371-93 



370-78 



Table XIV. — Mutual Inductances M 9 . 



Radius of circle. 


Radius of coil. 


Length of coil of 
192 turns. 


Distance of mean 

diametral plane of 

coil from circle. 


M 2 . . 


cm. 


cm. 


cm. 


cm. 




26-7870 


17' 


9419 


16' 





152 


1 


234 


•26 


26-7870 


17' 


9419 


16 





152 


•4 


232' 


94 


26-7870 


17' 


9369 


16 





152 


1 


234 


•14 


26-7370 


17* 


9419 


16 





152' 


1 


233 


'42 


26-7870 


17 


'9419 


16 





182 


•6 


137 


'86 


26-7870 


17 


•9419 


16 





182 


'9 


137 


•20 


26-7870 


17 


•9369 


16 


•o 


182 


•6 


137 


•79 


26-7370 


17-9419 


16-0 


182-6 


137-36 



78 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



Table XV. — Mutual Inductances M 9 . 



Distance apart of 

mean planes of 

coils. 



cm. 



Distance apart of 
contact circles. 



cm. 



Radius of contact 
circle 2. 



cm. 



Radius of coils. 



cm. 



M* 



Giving M 2 of Coils on Cylinders Nos. 1 and 2, with the Contact Circle of Disc 2. 



30-20 
30-50 
30-50 
30-50 
30-55 
30-60 
30-80 



167*50 


26- 


167-35 


26- 


167-65 


26- 


167-50 


26- 


167-50 


26- 


167-50 


26- 


167-50 


26- 



7800 
7800 
7800 
7800 
7800 
7800 
7800 



17 
17 
17 
17 
17 
17 
17 



9406 
9406 
9406 
9406 
9406 
9406 
9406 



370 
371 
369 
370 
370 
371 
371 



57 
89 
91 
90 
95 
01 
23 



Giving M 2 of Coils on Cylinders Nos. 3 and 4, with the Contact Circle of Disc 1. 



30 
30 
30 
30 
30 
30 
30 
30 



20 
50 
50 
50 
55 
60 
70 
80 



167 
167 
167 
167 
167 
167 
167 
167 



50 
35 
65 
50 
50 
50 
50 
50 



26 
26 
26 
26 
26 
26 
26 
26 



7910 
7910 
7910 
7910 
7910 
7910 
7910 
7910 



17 
17 
17 
17 
17 
17 
17 
17 



9430 
9430 
9430 
9430 
9430 
9430 
9430 
9430 



370 
372 
370 
371 
371 
371 
371 
371 



94 
27 
28 
27 
32 
38 
49 
60 



The change in mutual inductance for an increase in radius of any section of the 
coil is most readily calculated by finding dM/dA for the brush contact circle and a 
second circle of the same radius as the coil, the distance apart of the two circles 
varying over the length of the coil. 

Table XVI. 



d = distance 

apart of the 

circles. 


Radius of first circle = 26*787 cm. 
Radius of second circle = 


Change in M for 

change of 10/x 

in radius of 

smaller circle. 


cm. 
17*9141 


cm. 
17*9419 


cm. 
17*9697 


Y 


-M = 

' ,. . A 




cm. 
23*3150 
19-2851 
15-2573 
11-1679 
7*3135 


88*8782 
232*7922 


89*1121 
113-5449 
145-4984 
187*1127 
233*5633 


89*3456 
113-8412 
145*9000 
187*6662 
234*3348 


0*0084 
0-0107 
0*0145 
0*0199 
0-0278 



RESISTANCE BY A METHOD BASED ON THAT OF LOKENZ. 



79 



In Table XVI. we give the values of the mutual inductance of two such circles. 
The radius of one circle is 26787 cm. and the radius of the other circle varies from 
17*9141 cm. to 17*9697 cm. In column 5 of the table we give the change in M for a 
change in radius of the smaller circle of 10/x. 

The values in columns 1 and 5 were plotted and the resulting graph was employed, 
in conjunction with the values of the diameters used in plotting the conicality curves 
(fig. 12), to calculate the correction for the conicality of the coils. The method is so 
obvious that we need only give the results. 



Table XVII. — Corrections to M x for the Conicality of the Coils. 



Coils on cylinder 

No. 


Correction to Mi. 


Eing end of cylinder 
near disc. 


Eing end of cylinder 
away from disc. 


1 

2 
3 
4 


-0-589 
-0-571 
+ 0-295 
+ 0-200 


+ 0-942 
+ 0-494 
-0-326 
-0-153 



The mutual inductance of one coil and a disc circumference is about 30,000 cm., 
and when cylinder No. 1 is reversed in position a change of 1*5 in M b or 5 parts in 
100,000, is brought about by the conicality of the coils. 



Correction for Variation in Pitch. 

The graph of the difference measurements of a coil absolutely uniform in pitch is a 
straight line such as OP, fig. 17. Such a coil may be called a perfect coil and any 
short section of it a " perfect section." That there is a difference in the mutual 
inductance of a " perfect " section such as OAP and a circle, and an actual section 
such as OBP and the same circle, is easily 
seen. With the exception of the wires 
at O and P, every wire in the actual 
section is farther from the circle than the 
corresponding wires in the perfect section, 
and the mutual inductance of the former 
section will in consequence be the smaller. 





( 


V 
























t 


>y\ 


KC 


) 




A 




/ 












2b>u. 






tx_ — 








► 


■+*- 










_ ~A 












P 


\ ' 













































Fig. 17. 



If the periodic curve is symmetrical with respect to the " perfect " curve, the reverse 
is true for the next section, but the difference is not so great and hence there is not 
perfect compensation. As we proceed along a coil with such periodic variations the 



80 



MR F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



individual corrections are alternately plus and minus, and the total effect of these 
half-period elements depends on the difference of phase at the beginning and end 
as in the case of diffraction effects in light. 

To obtain the corrections we have calculated the mutual inductance of small 
portions of the coils and the corresponding circles. The mutual inductances calculated 
are (l) those of the circle and of coils each of which is 0*635 cm. in length, the 
distance between the circle and the end of the coil nearer to it increasing from 
7*3885 cm. to 23*2635 cm. in steps of 0*635 cm. ; (2) those of the circle and of coils, 
each 25/x long, the distance between the coil and circle increasing from 8*0235 cm. to 
23*2635 cm. in steps of 1*27 cm. In all cases the number of windings on a coil is at 
the rate of 12 turns per cm. The results of these calculations were plotted and by 
means of the curves we have corrected for the variation in pitch. 

As an illustration, let us examine the second section of coil No. 3 when the ring 
end of the cylinder is nearer to the disc. For clearness, this section is reproduced in 
fig. 17 and is denoted by CB. The end of the section nearer to the disc is 5/x nearer 
than the end of the corresponding perfect section, and the other end of the section is 
20/x farther from the disc than the other end of the corresponding perfect section. 
The ratio of the lengths of the two sections is therefore 6375/6350 and the ratio of 
the current densities in the equivalent current sheets is 6350/6375. The M of the 
perfect section and the disc is 1723*692 ; if its length is increased by 5/m at the end 
nearer the disc and by 20/* at its other end, M is increased by the amounts 1*382 and 
5*340. (These corrections are obtained from the curve.) The total M is now 
1730*414. If next we reduce the number of turns to its original value the M becomes 
1730*414 (6350/6375) = 1723*628, and since the section is now identical with the 
actual section, the correction for variation of pitch and displacement of section is 
1723*628-1723*692 = -0*064 or -3*7 parts in 100,000. 

In this way we have calculated the corrections for all the sections of the coils both 
when the ring end is nearer to the disc and when farther from it. 

The values of the corrections are as follows : — 

Table XVIII. 





Correction to M. 


Coils on cylinder 




i 


No. 


Eing end of cylinder Eing end of cylinder 
near disc. away from disc. 


1 

2 
3 
4 

1 


+ 0-502 
4-0-987 
-0-734 
-1-535 


-0-369 

-0-874 
+ 0-536 
+ 0-486 



EESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 



81 



Section 19.— Effect of Electbic Motor on the Mutual Inductance 

of the Coils and Discs. 

Originally it was our intention to employ a water turbine to drive the Lorenz 
apparatus, but there were many difficulties in the way of a satisfactory system, and 
finally we decided on an electric motor. Before coming to this decision we made a 
number of experiments on the effect of the presence of a large mass of soft iron on 
the mutual inductance of two coils each of 200 turns of wire and 24 cm. in diameter. 
The distance between the coils could be changed and the approximate mutual 
inductance between them was measured by Mr. A. Campbell. The iron employed was 
a mass built up of laminated sheets tightly pressed together, the dimensions of the 
mass being 35x18x8 cm. When the coils were parallel and their mean planes 7 cm. 
apart, the following values were obtained for the mutual inductance when the iron 
was placed on the common axis of the coils and at a distance d from their mean 
plane : — ■ 



d in centimetres. 


M in microhenries. 


00 


4942-5 


70 


4942*6 


60 


4942-6 


39 


4945-2 


20 


4975-0 


00 


4942-6 



The distance apart of the coils was afterwards increased to 10 cm. and eventually 
to 20 cm., but in no experiment could the effect of the iron at a distance of 1 m. be 
detected by the mutual inductance measurements. The stray magnetic field produced 
by the motor is not a source of trouble and has no influence on our results ; the only 
manner in which the motor can affect the resistance measurements is by its action, as 
a mass of iron, on the mutual inductance of the coils and discs. 

The field magnet of the motor is not large ; its length parallel to the shaft is about 
30 cm., and its section at right angles to the shaft is about 1000 sq. cm. The 
distance from the centre of the motor to the centre of the nearest of the four coils is 
almost exactly 400 cm. 

Dr. G. F. C. Searle, F.R.S., who has taken great interest in our work and to whom 
we tender our thanks, has very kindly calculated the effect of a sphere of soft iron on 
the mutual inductance of a coil and circle, both of which are some distance away. If 
the radius of the sphere is 20 cm. (corresponding roughly to a mass of iron equivalent 
to our motor) and the coil and circle are of the same dimensions and the same distance 
from the sphere as in the Lorenz apparatus, the effect of the sphere on the mutual 
inductance appears to be about 1 part in 10,000,000 which is, of course, absolutely 
negligible. This value is in very good agreement with two values determined 
experimentally. 

vol. ccxiv. — A. M 



82 MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OP A 

The intensity of the magnetic force, at a point corresponding to the centre of the 
motor, which is produced by a current of 2 amperes through the four coils of the 
Lorenz apparatus, is about 0'0025 C.G.S. units. The iron of the motor becomes 
magnetised and the result is an increase in the total magnetic flux through the discs. 
We had to find the ratio of this increase to the flux produced by the current in the 
four coils. This ratio we have found in two ways. In both of these we magnetised 
the iron of the motor by winding around it a large solenoid of 16 turns of insulated 
copper wire. When a current was passed through this coil the magnetising" field 
inside the solenoid was approximately in the same direction as the field due to the 
four coils, i.e., practically parallel to the shaft. The iron was magnetised and the two 
discs of the Lorenz apparatus were caused to rotate in the resulting field. Since the 
mean field in which one disc rotated was not the same as that in which the other disc 
rotated, a difference of potential was produced between the edges of the two discs. 
In our experiments we compared the galvanometer deflection produced by this potential 
difference with that produced by a current in the four coils of the Lorenz apparatus 
when the mutual inductance of the coils and discs was changed by a known amount. 
The result found is that when the iron of the motor is placed in a magnetising field of 
2*5 C.G.S. units, the total effective magnetic flux through the two discs is 1 part in 

' O O X 

10,000 of that due to the current in the four Lorenz coils. With such a small 
magnetising field as 0*0025 C.G.S. units, it was impossible to measure any effect, but 

the calculated effect is — ■, i.e.> 1 part in 10,000,000. 

In the second experiment we wound on a framework of wood a similar coil to that 
surrounding the motor. Both coils were of 20 cm. radius and had 16 complete turns. 
When a current of 5 amperes was passed through the coil, the intensity of the axial 
field was measured by means of a magnetometer at distances of 1, 2, 3, and 4 metres 
from the mean plane of the coil. Similar measurements were made along the axis of 
the coil which surrounded the motor, the current through this coil being 5 amperes as 
before. The results are of interest and are given in Table XIX. 

The nearer disc of the Lorenz apparatus is about 400 cm. from the centre of the 
motor, and we may take the magnetic force due to the motor as uniform over this 
disc. The area of one face of the disc is 2200 sq. cm. so that the total flux through 
it due to the iron of the motor was 2200 (0*0010-0*0003) = 1*5. Had the 
magnetising force on the iron of the motor been that due to 2 amperes through the 
four coils of the Lorenz apparatus, the total flux would be reduced to 
(1*54 x 0'0025)/2*5 = 0*0015. The total effective flux through the two discs will be 
somewhat less than this, but the reduction is not important. When making a 
resistance measurement, the total effective flux through the two discs is about 24,000, 
so that the magnitude of the motor effect is 0*0015/24000 = about 6 parts in 
100,000,000. 

The intensity of the stray field of the motor when running at full load has already 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 



83 



Table XIX. — Giving the Intensities of the Axial Magnetic Fields produced by a 
Current of 2 Amperes in Two Solenoids One of which has the Motor as a Core. 



Axial distance from 
centre of coil. 


Intensity of magnetic field. 


No iron present in coil. 


Motor as core of the coil. 


cm. 
100 
200 
300 
400 


C.G.S. units. 
0-023 
0-0034 
0-0009 
• 0003 


C.G.S. units. 
0-09 
0-013 
0-0036 
0-0010 



The motor was not running during these measurements. 

been stated to cause no trouble. The intensity of the component of this stray field 
parallel to the shaft of the apparatus and 400 cm. from the centre of the motor, was 
measured to be 0'0006 C.G.S. units. 

The general conclusion is that an electric motor may safely be used for such a 
purpose as ours. 

Section 20.— Arrangement of Circuits. 

(a) Connection to Brushes. — The brushes are connected to insulated terminals fixed 
to the phosphor-bronze rings, and double silk-covered copper connecting wires pass 
from these terminals to a selector switch on the observation table. A diagrammatic 
representation of the connections is given in fig. 19. The selector switch consists of 
two conducting arms which enable the observer to complete the circuit through any 
one of the five pairs of brushes by moving the contacts from stud to stud. When it is 
desired to place the ten brushes in series and thereby get an induced voltage five 
times as great as with one pair of brushes, conducting straps are placed in the 
positions SS SS indicated by the dotted lines and the turning head makes contact 
with the studs 1/ and 5. Any one of the five pairs of brushes may still be selected 
and observations may be made to test if the brush contacts are satisfactory. When 
the ten brushes are placed in series the thermo-electric effects are of course added 
and particular care has therefore to be taken with the brushes. When trouble is 
experienced it greatly facilitates the work to select pairs of brushes and so detect the 
faulty ones. 

To place the brushes in two sets of five in parallel, conducting straps connect 
together each set of five terminals TT, &c, on the selector switch. The position of 
the turning head is immaterial. 

(b) Multiple Commutator and Plug Board (fig. 19). — This is designed after the 
manner of the multiple commutator used for the Ayrton- Jones current weigher, and a 
short description will therefore suffice. A commutator allows of the reversal of the 

M 2 



84 MR F. E. SMITH ON THE ABSOLUTE MEASUEEMENTS OF A 

current in both coils on any one cylinder, and the plug board allows of the reversal of 
the current in any one or more of the helices. Each helix is designated by a number 
and a letter which are marked on an ebonite bridge at the top of the multiple 
commutator ; the turning heads are also numbered to enable changes to be rapidly 
made without likelihood of error. The lower commutator reverses the current in all 
of the coils. 

By suitable conducting straps the coils on one cylinder may be placed in parallel 
with those on another cylinder. In many of our observations the coils on cylinders 
1 and 2 were placed in parallel with the coils on cylinders 3 and 4. 

(c) The Reversing Switch. — Preliminary experiments indicated that the make and 
break of the current through the coils had a considerable inductive action on the 
galvanometer circuit. The deflection of the galvanometer thus produced amounted 
at times to 50 cm., on a scale at a distance of 2 m., and prohibited such rapid 
reversal of the current as we desired to make. Similar inductive action is remarked 
on by Prof. J. V. Jones in the 'Report of the British Association for 1890/ 
Had the galvanometer circuit been a stationary one, we might have introduced a 
compensating system, but this was not possible with the system of rotating conductors 
we had installed. It was evident that the galvanometer circuit should be broken 
before making or breaking the current circuit, and as the position of rest of the 
galvanometer coil on open circuit is, in general, different from that when the circuit is 
closed, it appeared desirable to have the galvanometer excessively damped when not 
included in the main circuit. The switch shown in fig. 19 enables this to be done. 
All the connections are shown in the figure and it is not necessary to describe the 
switch in detail. On moving from stud 1 to stud 2 the galvanometer circuit is 
shunted by a negligible resistance and the main galvanometer circuit is broken. On 
moving to stud 3 the current is broken— on to stud 4 the current is made again but 
in the reverse direction—and when the movement is continued to stud 5 the 
galvanometer is again placed in the brush circuit. This switch proved very 
convenient in practice and greatly facilitated our work. 

(d) In addition to the commutators already described, a simple commutator was 
added to reverse the potential leads attached to the standard resistance. Such 
reversals were made to eliminate electrostatic effects. 

(e) Galvanometer. — This was of the Ayrton-Mather type and was very kindly lent 
to us for the work by The Cambridge Scientific Instrument Company, Limited. The 
resistance of the galvanometer is 16 '5 ohms and the period of the coil is 5*2 seconds. 
At a distance of 2000 scale divisions the sensitiveness is 57 divisions per microvolt. 
The external resistance for aperiodic working is 50 ohms. 

When the current through the eight coils of the Lorenz apparatus is 2 amperes 
and a direct measurement of a resistance of 0*01 ohm is in progress, the difference of 
potential on the standard resistance is 0'02 volts. In making the measurements the 
current is reversed, and on reversal there is a change in the rest point of the 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 



85 



C 



R 

AAAAAt 



<S> 



B 



~-J 



Fig. 18. Simple diagram of circuit. 
R, resistance to be measured ; B, battery ; Gr, galvanometer. 



TO BRUSHES 

ON DISC I 

TO BRUSHES 

ON Dl SC 2 



i i i i i 

12 3 4-5 
9 6 O^O.OT 



fO 



(f) 

or 
Q 

z 

-J 

> 

o 

z 

o 



if) <\l 

5 
o 



dddiS 



O 




Fig. 19. Detailed diagram of circuit. 
A, ammeter ; B, battery ; C, brush selector switch ; D, resistance in galvanometer circuit ; E, earthed 
point on circuit; G, galvanometer; M, multiple commutator and plug board; E, resistance to be 
measured ; S, reversing switch. 



86 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OP A 



galvanometer spot of 57 mm. (at 2 m.) per microvolt. To obtain a precision of 1 part 
in 100,000 on a single reading it is necessary, therefore, to note the change of 
deflection when the current is reversed with an error not greater than 11*4 mm. 
When the motion of the coil is made aperiodic by introducing additional external 
resistance, the sensitiveness is about one-quarter of the previous value, and the error 
of the difference reading must not be greater than about 3 mm. For measurements 
of a resistance of 0*001 ohm the difference of potential on the standard resistance was 
about 0*004 volt and the change of deflection had to be read with an error not greater 
than 0*6 mm. For a single measurement of a resistance about fifty different readings 
were taken and the mean of these is used to calculate the result (see p. 98). 

Fig. 18 is a simple diagram of the circuit and requires no explanation. Fig. 19 is 
a more detailed diagram and shows the connection of the circuit to earth. 

Section 21.— Standard Kesistances. 

The resistances measured in absolute measure were of nominal values 0*001 ohm, 
0*002 ohm, and 0*01 ohm. The first and last of these were standard manganin 
resistances capable of carrying currents of 30 and 10 amperes respectively without 

increasing in temperature by an appreciable amount. 
The currents used in our measurements did not exceed 
4 and 2 amperes ' respectively, and the heating effect 
due to these currents could not be detected. 

To obtain an effect corresponding to that of a resist- 
ance of 0*002 ohm, three standard resistances were 
arranged in a triangular fashion (fig. 20) as first 
suggested by Lord Bayleigh. # The resistances 
are of the well-known type introduced by the 
Physikalisch-Technische Reichsanstalt ; they are of 
manganin and the mean temperature coefficient in the 
neighbourhood of 20° C. is about + 15 parts in 1,000,000 
per degree rise of temperature. The resistance (a) fig. 20 is a coil of 2 ohms 
resistance, (6) is a 1-ohm standard, and (c) consists of a coil of 1000 ohms shunted by 
others having values from 100,000 to 500,000 ohms. If the current in the main 

circuit is i the current through b is =— — =- where L is the summed value of the 

a + b + c + L 

resistances of the current leads to the three coils. The difference of potential at the 

9 The quantity =— — =- thus takes the place of R 

a+b+c+L a + b + c + L 

in the formula R = M.n (p. 35) and is called the effective resistance. f The two ohm 

* 'Phil. Trans./ 1883. 

t A complete treatment on the combinations of a four-terminal resistance with other resistances is 
given by G-. F. C. Searle, £ The Electrician/ March 31 to April 21, 1911. 




extremities of b is 



EESISTANGE BY A METHOD BASED ON THAT OF LORENZ. 87 

coil is of special construction ; it is divided into two equal coils each of which can be 
compared with the standard coils. 

The coils were supported from mercury cups in a bath of well stirred paraffin oil 
maintained at a constant temperature of 20°'0 0. The stirring was produced by 
blowing dry air through the oil. A spiral toluene thermostat was used to control the 
temperature and the heating of the oil was produced by an electric current flowing 
through a resistance coil supported on a large frame at the bottom of the bath. 
These arrangements are those in common use in the Electrical Standards Department, 
and the results obtained are exceedingly satisfactory. 

The methods adopted for the accurate comparison of the coils with other wire 
standards and with mercury standards of resistance are published elsewhere, # and it 
is not necessary to describe them here. All the resistances were frequently compared 
with standard manganin coils which are hermetically sealed and the secular changes 
in which are exceedingly small. 

♦The new mercury standards of resistance of the National Physical Laboratory are 
nine in number. They have spherical end vessels 4 cm. in diameter and were made in 
accordance with the specificationf of the London Conference on Electrical Units and 
Standards 1908. The resolution relating to the international ohm is as follows : — 

" The international ohm is the resistance offered to an unvarying electric current 
by a column of mercury at the temperature of melting ice, 14*4521 gr. in mass, of a 
constant cross- sectional area and of a length of 106*300 cm." 

Mercury standards of resistance have also been made at the Physikalisch-Technische 
Reichsanstalt and the Bureau of Standards at Washington. Recent comparisons 
show that the units of resistance so derived agree with the unit derived at the 
National Physical Laboratory within about 2 parts in 100,000. The exact figures 
are not yet to hand. 

Section 22. — Setting of the Coils to be Coaxial with the Shaft and at 
approximately equal distances from the brush contact circles. 

(l) Setting of the Axes of the Coils to he Parallel to the Axis of the Shaft. 

Mechanical Method. — -At the time of turning the marble cylinders, a phosphor- 
bronze ring was let into one end of each, and the surface turned at right angles to 
the axis. When the cylinders were in position on the cradles of the Lorenz 
apparatus, a radial arm supporting a micrometer head was clamped in a suitable 
position on the shaft and contact between the micrometer screw and each of the rings 
in turn was made at three points 120 degrees apart. The making of a contact was 

* 'B.A. Elect. Stands. Committee Report/ 1906. 
t 'B.A. Elect Stands. Committee Eeport/ 1909. 



88 MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

indicated by a telephone, and readings could be repeated within about jO'Ol mm. 
Each cylinder was adjusted in position until the three readings were identical and the 
axes of the cylinders were then approximately parallel to the axis of the shaft. The 
method was sensitive, but irregularities in the thrust-bearing surface of the shaft 
must introduce errors which are difficult to eliminate. 

(2) Setting of the Coils to be Coaxial with the Shaft. 

(a) Mechanical Method. — A direct-reading spring indicator was used to measure 
the perpendicular distance of the inner surface of each marble cylinder from the 
surface of the shaft, and each cylinder was adjusted in position until the readings at 
all points were practically identical. The indicator was sensitive to a difference of 
about 0'02 mm. and the adjustment of a cylinder occupied only a few minutes. If 
the inner surface of a cylinder is not coaxial with the coil, then of course an error is 
introduced. 

(b) Electrical Method. — -The calculations we have made give only the mutual 
inductance of the coils and the brush contact circles, and it is not possible for us to 
give other than general diagrams indicating the manner in which the intensity of the 
magnetic field varies along the radius of a circle and other diagrams indicating the 
difference of potential between the axis and any point on a radius which is produced 
by the rotation of that radius. Such diagrams will, however, serve to explain the 
electrical method of setting the coils. 

In fig. 21 (a), DC represents the plane of rotation of one of the discs, and BB' the 
position of the brush contacts. The intensity of the magnetic field produced by a current 
in the coils on both sides of the disc is practically zero at BB' and changes in sign as 
we pass radially outwards from the disc. The form of the intensity curve is roughly 
shown by the curve EFGH. When the disc rotates, the difference of potential 
between the centre O and any point A on a radius increases with increase of the 
distance A to a maximum value at B and B', but afterwards it decreases. The 

potential difference is given by 2irn ~Bydy where OA = y and B is the magnetic flux 

Jo 

at a distance y from O. The potential difference can be represented as an area if the 
flux be multiplied by y before it is plotted. The area enclosed by the line DD' and 
the curve KCLM shows roughly how the potential difference varies ; thus the 
difference of potential between and A is represented by the area OCA. Areas to 
the left of DD' are counted as positive and those to the right are counted as negative. 
In our apparatus the brush and segment at B are insulated from those at B' and it is 
therefore possible to measure the difference of potential between the segments and so 
determine whether the field produced by the current in the coil is symmetrical with 
respect to the axis of the shaft. Without the insulation of the segments, i.e., by the 
use of a disc alone as in the old forms of apparatus, eddy currents in the disc produce 
disturbing effects. 



RESISTANCE BY A METHOD BASED ON THAT OP LORENZ. 



89 



The effect of displacing the coils parallel to the shaft is diagrammatically shown in 
fig. 21 (6). The area OB'CS is greater than the area OLR — RNB and the difference 
of potential between O and B' is greater than that between O and B. The relation 
between the mean of these voltages and that obtained when the coils and shaft are 
coaxial will be seen presently. 

A similar change in the difference of potential between B and B ; is produced by a 
rotation of a coil about a vertical axis passing through its centre. Fig. 21 (c) shows 




D 




(') 



Fig. 21. 



generally the effect produced and makes it clear that the mean diametral plane of the 
coil may be set parallel to the brush contact circle by observations of the differences 
of potential at such points as BB'. 

The first coils to be set coaxial with the shaft by these electrical methods were 
those on cylinders Nos. 1 and 4. Afterwards the mutual inductance of the No. 1 
coils and the brush contact circles was made equal to that of the No. 4 coils and the 
same circles by passing a current in the same direction through both and altering the 
distance between the No. 1 coils and the disc nearer to them until on reversal of the 
current in the coils there was no change in the difference of potential between the two 
contact circles. The object of this procedure was to enable us to measure directly 
the amount by which a potential difference such as OB'CS, fig. 21 (&), exceeded 
the normal potential difference such as OB'C, fig. 21 (a). The circuit formed will be 

VOL. CCXIV.— A. N" 



90 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



clear from the small diagram at the top of fig. 22. One brush made contact with 
one (or two) segments on each disc and the circuit was completed through the rotating 
conducting wires. The displacements were read to 0*001 mm. and the changes in 




-^300 



-f-200 — 



< 



uJ + iOO 
> 

u ^ 



< 
V 



oo 



200 



300 



-400 















































































































































































A 






































































































































































































B 




















































































































































/ 































































































































































IO 



e 



6 4 2 O 

DISPLACEMENT IN 



2 4 6 

MILL! MET RES 



8 



Fig. 22. 



4- i5 



•4- IO 



+ 5 



O 



o 



CD 

id 
> 

o: 



UJ 

< 

U 
if) 



ZQ 



IO 



A, change (parts in 100,000) in voltage induced by rotation of radius E due to displacement of 
cylinder in direction perpendicular to its axis; B, change in mean voltage induced when five 
brushes are used, touching points 72 degrees apart on edge of disc. 



potential were calculated from the changes in the galvanometer deflection which were 
produced on reversal of the current in the coils. 

A displacement of 10 mm. increases the difference of potential between O and B' ? 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 



91 



fig. 21 (a) and (6), by about 843 parts in 100,000, and diminishes that between* O and 
B, fig. 21 (a) and (6) by about 37 1 parts in 100,000. If one brush only were employed 
on the segments during measurements of a resistance it would therefore be necessary 
to ensure coincidence of axes of coils and shaft within about j-^u °f a millimetre, in 
order that the error associated with the setting should not exceed 1 part in 100,000. 




UJ 

> 

o 



UJ 

j 
< 





+ 200 


■ 


B ** 






■ 














— — -M---1 














■Auatfaattw 












































+ IOO 














^\ 


































































— — — . 


o 
















































































-IOO 














"B 














































































































-20O 










































-300 






































\ 

































































































































+ 40 



+ 20 



O 



- 20 



~40 



CO 

UJ 

> 

D 
U 



< 
U 



-GO 



-+ 



5° £° i° o f a° 3 1 

ROTATION ABOUT VERTICAL AXIS 



4-' 



5 



Fig. 23. 

A, change (parts in 100,000) in voltage induced by rotation of radius R due to angular displacement 
of cylinder about vertical axis passing through mean diametral plane ; B, change in mean voltage 
induced when five brushes are used, touching points on edge of disc 72 degrees apart, 



In practice, however, five brushes are used on each disc, and when these are used in 
series with the five brushes on the other disc, a displacement of 10 mm. diminishes 
the voltage by (371 — 343)/2 = 14 parts only in 100,000. When the brushes on each 
disc are placed in parallel the mean potential difference measured by the galvanometer 

N 2 



92 MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

is nearly the mean of the separate potential differences and the error of setting is 
about the same as when the brushes are placed in series. With the brushes in parallel 
it is better, however, to include a resistance in series with each brush before connecting 
the brushes together, for then any difference of resistance between the brush contacts 
is rendered negligibly small. The observed differences of potential due to displacements 
when the brushes were placed in parallel are plotted in fig. 22, and the equation of the 
resulting curve is y — 0'132^ 2 where y is the change in voltage in parts in 100,000 
produced by a displacement of x mm. from the coaxial position. An identical curve 
results when the brushes are placed in series. It is clear that the observations enable 
the coil and shaft to be set coaxial within less than 0*1 mm. and an error of this 
amount introduces an error in the resistance measurements of about I part in 
100,000,000. 

Similar observations were made when one of the cylinders was rotated about a 
vertical axis P. The results are plotted in fig. 23. The equation when the brushes 
are placed in series or in parallel is y = S'5x 2 where y is the change in voltage in parts 
in 100,000 produced by a rotation of x°. Thus, if x = 0*1° (a large amount) the error 
introduced is 4 parts in 10,000,000. 



(3) Setting of the Coils on opposite Sides of a Disc to be at the same Mean 
Distance from the Contact Circle of Segments and Brushes, 

The current through the coils on cylinder No. 1 was caused to circulate in the 
same direction as the current through the coils on cylinder No. 4, and the resulting 
flux through disc No. 1 was therefore in the same direction as that through disc No. 2. 
The galvanometer circuit was completed without the inclusion of the standard 
resistance R, and reversals of the current through the coils were made in the usual 
manner to eliminate all effects but that due to a difference of flux through the discs, 
caused by the current. In general there was a want of balance and in such case the 
distance of one of the coils was changed until balance was secured or very nearly 
so. With care equality could be obtained within about 2 parts in 100,000, but in 
general we were content to ensure equality within about 25 parts in 100,000. This 
is equivalent to equality, within about 0'07 mm. of the mean distance of the No. 1 
coils from No. 1 disc, and that of the No. 4 coils from, the No. 2 disc. A similar 
setting of the No. 2 and No. 3 coils was next made, and afterwards the current 
through coils Nos. 1 and 3 was made to circulate in the opposite direction to that in 
coils Nos. 2 and 4. If all the coils are at equal distances from the discs, there is 
now no deflection of the galvanometer and the positions of coils Nos. 1 and 4 or of 
Nos. 2 and 3 are changed by equal amounts until a balance is secured. The coils on 
all of the cylinders are thus placed at the same mean distance, within about 0'1 mm., 
from the brush contact circles and. such equality is sufficient. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 



93 



Section 23.-— Determination of Eesistanoe in Absolute Measure. 

(l) Preliminary Tests. 

(A) Elimination of Error due to the Thermo-electric Effects at the Brush Con- 
tacts. — -The thermo-electric effects are eliminated by observing only the change of 
deflection of the galvanometer with reversal of the current in. the coils. If the 
magnitude of the changes of the thermo-electric effects is considerable, and great 
accuracy is desired, a large number of reversals are necessary. It becomes important, 
therefore, to determine the general magnitude of the error introduced by making a 
comparatively small number of reversals. 




o 
> 

o 

q: 
u 



O 



3 



9 



IO 



5 6 7 8 

TIME IN MINUTES 

Fig. 24. 
A, variations in thermo-electric effects ; B, variations in thermo-electric effects plus variations in speed. 

Our normal procedure in making resistance measurements was to reverse the current 
in the coils at intervals of 15 seconds, and in practically all cases 48 reversals were 
made. To determine to what extent this procedure eliminated the thermo-electric 
effects, we operated the reversing switch with no current flowing in the coils and 
noted the galvanometer deflection at the end of every 15 seconds. 

The differences obtained in one such experiment are plotted against time in graph 
(A) fig. 24. The mean difference from the mean is ± 0'03 mm., which would intro- 
duce an error in a measurement of resistance of something less than. 1 part in 



94 MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

1,000,000. The mean difference from the mean irrespective of sign is ±0'8 mm. 
These differences are typical of those obtained in our experiments and from them we 
conclude that the thermo-electric effects introduce no error in our final result. 

(B) Elimination of Error due to Electrostatic Effects. — -We have already described 
the connections of the circuit to earth and the reversals made in order to eliminate 
any electrostatic effect on the galvanometer, but we thought it desirable to make 
tests from time to time of the efficiency of our arrangements. In these experiments 
the coils were removed from the circuit and an equivalent resistance substituted, 
which resistance was placed in such a position that the current through it produced 
no magnetic field in the neighbourhood of" the discs. The standard resistance 
remained in the circuit, but the leads connecting it to the galvanometer were 
connected to one and not both of the potential terminals. Observations of the change 
of deflection were made as before at intervals of 15 seconds. The mean difference 
from the mean of such a set of measurements was always negligibly small and 
indicated that no error due to electrostatic effects could influence our results. 

(C) Constancy of the Speed. — -The records of speed are remarkably good and show 
the variations to be very small. Each record enables the mean speed, during the 
measurement of resistance associated with it, to be calculated with great accuracy ; 
but a fluctuation in speed of a few parts in 100,000 lasting for a comparatively short 
time cannot be detected. The metjiod adopted to control the speed is described in 
Section 10, and it is evident that the speed must oscillate in value, the magnitude of 
the oscillations depending on the cause of the variations, the sensitiveness of the 
bridge, the sluggish movement of the galvanometer, and the ability of the 
operator. Now the galvanometer in the bridge circuit is sluggish in its movements 
and unsuited to detect changes in speed lasting for a few seconds only, but the 
Cambridge galvanometer used in the Lorenz circuit has a short period (5*2 seconds), 
and the changes in its deflection during a measurement of resistance show that 
variations in speed lasting for a few seconds only are of common occurrence. 

An idea of the magnitude of these sudden changes of speed is afforded by a com- 
parison of the two graphs (A) and (B), fig. 24. Graph (A) shows the variation in 
the thermo-electric effects, while graph (B) shows the variation in the thermo-electric 
effects plus the variations in speed. The differences plotted in graph (B) are those 
taken during a measurement of a resistance. These two graphs are typical of the 
results generally obtained. In all our experiments the combined variation of 
thermo-electric effects and changes of speed were of about twice the magnitude of the 
changes in thermo-electric effects alone, and it appears reasonable to conclude that 
small oscillations in the speed were frequent. The magnitude of these oscillations 
appears to be about 5 parts in 100,000. However, such fluctuations had practically 
no effect on the final results. 

(D) Effect of the Leads.— -It was possible that the current in the leads to and from 
the coils might produce a magnetic field of sufficient intensity to affect the results. 



RESISTANCE BY A METHOD BASED ON THAT OF LOEENZ. 95 

This was tested by completing the main circuit through the leads only (omitting the 
coils) and taking a few complete sets of observations. No effect could be measured 
and we conclude that the current in the concentric leads and remaining portions of the 
circuit other than the coils have no influence on our results. 

(2) Normal Procedure in making an Absolute Measurement of a Resistance. — In 
making a determination of a resistance in absolute measure, we determined (a) the 
insulation resistance of the various parts of the circuits ; (b) the mean distance between 
the mean diametral planes of the coils ; (c) the temperature of the marble cylinders ; 
(d) the want of equality between the product of speed and mutual inductance and the 
resistance, the value of which was desired ; (e) the temperature of the marble 
cylinder ; and (f) repetition of (6). 

To measure (b) and (/) we commenced with observations on the invar line standard ; 
we then observed the eight plug distances and concluded with further readings on the 
invar. These observations lasted about twenty minutes. 

The temperatures (c) and (e) of the cylinders were taken by means of thermometers, 
and the effective temperature was calculated by means of the formula given in 
Section 13. In general, the difference of temperature (e) — (c) was about 2°*2 C. To 
determine (d) the commutators and plugs were correctly set, the galvanometer circuit 
closed, and the reversals of current, &c, made which are indicated in Section 20. At 
the same time the speed was maintained constant Avithin a few parts in 100,000 and a 
record of the speed taken. The time occupied was usually from 17 to 20 minutes. In 
general, the first 12 reversals of current produced changes in the deflection due to a 
difference Mn— R where R is the value of the standard resistance in absolute measure. 
The next 8 reversals were made when R was reduced in value by 1 part in 1000 by 
shunting it with another standard resistance. Then followed 24 more readings with 
R unshunted; 8 with R shunted, and the final 12 measurements were made with R 
again normal. The sensitiveness of the arrangement is, of course, directly proportional 
to the current through the coils, and as this was not constant from day to day we 
thought it best to determine it on every occasion. 

Before proceeding with a measurement, the brushes were wiped with wash-leather 
and the tension on the wires adjusted. The variations in the thermo-electric effects 
were then observed and the petrol lubrication attended to. Bad lubrication on disc 
No. 1 produced a deflection of the galvanometer in one direction, and bad lubrication 
on disc No. 2 produced a deflection in the reverse direction. With a good supply of 
petrol on the edges of the discs there was practically no deflection when the brushes 
were in good condition. When the brushes were somewhat worn— usually after 
6 runs — -it was not possible to take good observations, and rather than waste time in 
making indifferent measurements we preferred to wait until new brushes had been 
inserted. Careful tests of the brushes were regularly made, and it is no doubt due to 
this fact that we are able to record the result of every completed measurement. At 
times the thermo-electric and speed variation effects were a little troublesome, but we 



96 ME, P. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

never considered them sufficiently serious to justify us in discarding the results. It is 
true that in two experiments we were not able to calculate the resistance ; in the first 
of these, the apparatus for recording the speed was not put in gear and so no record 
was made ; in the second, an interference with the battery connected to the motor 
produced a change of speed beyond our control and we had to abandon the experiment. 
In all other cases when a set of measurements was commenced, it was completed and 
the result is given in this paper. 

At a speed of about 17 '4 revolutions per second, the product Mn is nearly 2 x 10 6 , 
and the corresponding resistance is therefore about 0'002 ohm. We have already 
described the combination of coils which gives such an effective resistance, and we 
used the apparatus for its measurement on ten occasions. 

To measure a resistance of 0*001 ohm, the coils on cylinders Nos. 1 and 2 were placed 
in parallel with those on cylinders Nos. 3 and 4. The latter coils were found to be 
equal in resistance to the former within, the limits of the errors of the measurements 
made, and a division of the main current into two parts, equal within about 1 in 3000 
could be ensured. In general, equality of the divided currents, within 1 per cent, 
would have been sufficient. The current through the standard resistance is now twice 
the mean value of the current through the coils and the resulting equation is 
iMn = 2iR. When Mn — 2xlO Q , R is 0*001 ohm. This arrangement was most 
convenient. A resistance of 0*001 ohm is more readily compared with 1-ohm standards 
than a resistance of 0*002 ohm, and as our standard of 0*001 ohm was of thick 
manganin strip, a current of 4 amperes could be passed through it without an 
appreciable heating effect. 

When the brushes are placed in series, a resistance of 0*01 ohm can be measured, as 
the equation 5iMn = iR then holds good. Although this arrangement is very sensitive, 
a slightly greater pressure is required on each brush and this frequently produced 
trouble. 

A sample series of readings, taken on June 7, 1913, gives a good idea of the 
measurements involved in a single determination :— 

June 7, 1913. 

(l) Observations on invar line standard. Temperature = 15° '5 O. Lines 4-616. 

Length at 15°'5 C. = 30'6016 7 cm. 

Microscope readings (corrected). 



Taken before readings on plugs 



Taken after readings on plugs 



Left. 


Right. 


Diff. (L - E). 


1154-9 


946-6 


+ 208-3/X 


1135-0 


1028-0 


+ 207-0 


1133-4 


1025-4 


+ 208-0 


979-4 


770-9 


+ 208-5 


982-4 


774-1 


+ 208 ' 3 


1035-1 


825-2 


+ 209-9 



- Mean = + 208 * 3/x. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 



97 



(2) Observations on plugs. 



Microscope readings (corrected). 



Plug. 

(7) 

(8) 
(4) 

(3) 
(10) 

(9) 
(13) 

(14) 



Left. 


Plug. 


927-3 


(16) 


1551-1 


(15) 


1052-0 


(11) 


328-9 


(12) 


1042-6 


(1) 


1167-6 


(2) 


294-3 


(6) 


1409-9 


(5) 



Right. 

179-2 
834-1 
1185-2 
2447-9 
1232-3 
1126-0 
1199-8 
757-5 



Diff. (L - R). 

748-1 
717-0 

- 133-2 
-2119-0 

- 189-7 

41-6 

- 905-5 
652-4 



Corr.,* d. 

- 750 

- 716 
+ 107 
+ 2094 
+ 177 

- 55-5 
+ 887 

- 669 



Mean 



Mean distance between the mean diametral planes of the coils 



= 30-6016 7 cm. -14-2/X-208-3/X - 30*5794 2 cm. 



(3) Temperature of Marble Cylinders. 



(L - R) - d. 



In 


Before resistance 


After resistance 


<IO. 


measurements. 


measurements. 




°C. 


°C. 


1 


16-6 


18-8 


2 


16-55 


18*75 


3 


16-5 


18-8 


4 


16*55 


18-9 



iffer 


enc 





C. 


+ 2 


•2 


+ 2 


•2 


+ 2 


•3 


+ 2 


•35 



Mean . . . 16*5* 



Mean . . . +2*2> 



- 1 
+ 1 
-26 
-25 
-12 
-13 
-18 
-16 



9/* 


2 

7 
9 
5 
6 



-14-2/x 



.-. Effective temperature = 16°-5 5 C. + 0-42 (2° -25 C.) = 17° -5 0. 



* The correction is equal to the difference of the distances of the plugs from the mean diametral planes 
of the coils, these distances being deduced from the metrology measurements. Thus we obtained from 
these measurements: — -Distance (di) of plug 7 from centre = 9*3892 cm., distance (d 2 ) of plug 16 from 
centre = 9*3142 cm. Hence, when the coils are parallel, the distance of plug 16 from plug 7 is greater 
than the distance between the mean diametral planes of the coils by 9*3892 - 9*3142 = 750/x and a 
corresponding correction must therefore be applied. For our purpose it is necessary to measure four 
distances between four pairs of plugs, but, in practice, we measured eight distances between eight pairs of 
plugs in order to obtain a check. The agreement was always good. 

VOL, COXIV. — A. O 



98 



ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



(4) Changes in the Deflection of the Galvanometer produced on Reversal of the 

Current in the Coils. 



Set No, 1. 


Sensitiveness 
(a). 

mm. 




St 


mm. 


mi 


+ 4" 


2 


-36-4 - 


+ o- 


2 


4' 


2 


40-0 


4' 


4 


1 





38-6 


o- 


2 


4 


•4 


38-0 


0- 


6 


2 


•8 


41-0 


6 





2 


•2 


37-0 


2 


•8 


7 


•0 


40-0 


. 7 


•0 


5 


■4 


36-2 


5 


•0 


1 


•6 


. 


4 


•0 


4 


•4 




4 


■4 


6 


•0 




1 


•0 


4-0 




1-0 



Set No. 2. 



mm. 



+ 6 
3 
1 

3 
3 
2 
1 
3 
2 
1 
4 





8 
2 


8 
4 
8 
8 
4 
4 
2 



Sensitiveness 


Set No. 3. 


mm. 


mm. 


-38' 


4 


+ 2- 


2 


38' 


4 


4- 


2 


36 


6 


4' 





40 


•2 


3- 


6 


41 


•0 


1 


4 


36 


•0 


2 


6 


37 


•2 








36 


•2 


1 


•0 


* 


3 


•8 




5 


•0 




4 


•4 






3-8 



Miean of Sets 1, 2, 3 = +3*2 mm. 

Mean of Sets (a) and (b) = - 38 • 2 mm. 

.*. Change of deflection for a diminution of the resistance of 100 parts in 100,000 

= -38-2-3-2 = 41-4mm. 
.-. The resistance is greater than the product Mn by — = 4-7*7 parts in 100,000. 

(4) Sets of readings similar to (l) and (2) on the invar standard and the plugs were 
made after the resistance measurements, hut these need not be given here. The mean 
distance obtained is identical within 1/x with that already given. Such remarkable 
agreement was in general found between the two sets of measurements that we 
conclude the expansion of the cylinders does not affect the position of the mean 
diametral planes of the coils. In such a case, with uniform expansion of the four 
cylinders, the plug distances keep absolutely constant. Because of this we did not 
always take a second set of readings. Constancy over seven or eight hours was, 
however, rarely obtained, for the bed of the machine usually increased in temperature, 
and its expansion resulted in a change in the distance between the cylinders. Of 
course, the distance was frequently changed intentionally to alter the value of M. 

(5) The temperature of the discs was about 17° 0. and the distance between the 
brush contact circles was 167'5 cm. The speed was found from the record to be 
17'3835 7 revolutions per second. 

(6) Calculation of Mn. : — • 

(a) The value of M x is 115893*7 when the temperature of the coils and discs is 
20° C. and the distance between the mean diametral planes of the coils is 30*5785 cm. 

(6) The value of M 2 is 742*4 for a distance between the brush contact circles of 
167*50 cm. 



KESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 99 

(c) When the distance between the diametral planes is increased to 30*5794 2 cm., 
M x is diminished by 3*19 ; and when the temperature of the coils is diminished to 
17°*5 C, M x is reduced by 3*06. 

(d) When the temperature of the disc is reduced to 17° C, M l — M 2 is increased 
by 012. 

(e) Hence the value of M is 

M = Mx-Ma = 115893*7-742'4-3*17-3*06 + 0*12 

= 115145*2 cm, 

(/) M.n is therefore 115145'2 x 17*3835 7 = 200163 5 . 

(7) Calculation of R«: — 
We have 



Hence 



iMn = 2iK{l +(7'7 x 10~ 5 )}. 

E = 200163 5 /2 + 7 7 
= 100089 5 cm./sec. 
= 0*00100089 5 ohm. 

This value of E is on the assumption that the position of the mean diametral planes 
has been correctly estimated from the metrology measurements. To remove the 
assumption it is necessary to reverse the coils without interchanging, and this was done 
after 28 measurements of the 0*001 ohm standard had been made. The effect of 
reversal is very small and indicates that the assumed mean position of the diametral 
planes is very nearly correct. The mean result obtained in the two positions is taken 
as the value of K. When the coils Nos. 1 and 2 are used independently of Nos. 3 
and 4, an appreciable difference— about 4 parts in 100,000 — is obtained on reversal of 
the coils. The conclusion is that the estimated position of the mean diametral planes 
of the coils Nos. 1 and 2 is incorrect by about 0*006 mm. The same is true for the 
coils Nos. 3 and 4. 

In Tables XX. and XXIII. we give the data relating to measurements of a resistance 
of nominal value 0*001 ohm, and Tables XXI. and XXII. give the results only of 
the measurements of a resistance of 0*01 ohm and an effective resistance of 0*002 ohm. 

The observed values in absolute measure and the values in international ohms (new 
N.P.L. Mercury Standards of Resistance) are given in Table XXIV. 

Probable Errors. 

The mean observational error of the results given in Table XX. is about 2 parts in 
100,000 for a single observation. This error includes all the errors arising from an 

* 

inaccurate estimation of the distance apart of the coils, of variations in the speed, of 
an erroneous estimate of the mean speed, of faulty temperature observations, and the 

o 2 



100 



ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 





Di = Mean distance between the mean diametral planes of coils 1 and 2 and of coils 3 and 4. 


M = Mutual inductance between the coils and the discs. 
n = Number of revolutions per second. 


E = Value of resistance in absolute measure, as deduced from the observations. 


(E > Mn) = Value deduced from mean change in galvanometer deflection on reversal of current. 


The values given for Di, M, &c, are not quite correct. They are subject to small corrections owing to the exact positions of the mean diametral 
planes of the coils not being known. These corrections are of exactly the same magnitude in Part B of the table, but they are of opposite sign. The 
mean value of E (Parts A and B) is the true one. 


The distance between the brush-contact circles was practically constant and equal to 167 *5 cm. 


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102 



MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 



Table XXI. — -Results of Measurements of 0*01 ohm at 20°'0 C. 



13.2.13 

4.3.13 

28.3.13 

29.3.13 

31.3.13 

31.3.13 

31.3.13 

1.4.13 

1.4.13 

3.4.13 

4.4.13 

5.4.13 



Mean 



Part A (rings away) 






Part B (rings near). 




Measured 


Difference from 




Measured 


Difference from 


value of R in 


mean. Parts 


Date. 


value of R in 


mean. Parts 


cm./sec. 


in 100,000. 
+ T8 


18.6.13 


cm./sec. 


in 100,000. 


100092 20 


100093 2O 


+ 3- 


5 


90 40 


±0* 





19.6.13 


87eo 


-1- 


9 


88 3 o 


-2' 


1 


19.6.13 


90 90 


+ 1' 


2 


90 50 


+ 0' 


1 


19.6.13 


91so 


+ r 


8 


93 20 


+ 2' 


8 


20.6.13 


8920 


-0' 


5 


88eo 


-1' 


8 


20.6.13 


8880 


-0' 


9 


93 40 


+ 3' 





21.6.13 


91 6 o 


+ i* 


8 


93 o 


+ 2 


•6 


24.6.13 


90 20 


4-0' 


5 


9190 


+ 1 


•5 


25.6.13 


86 70 


-3 


•0 


8850 


-1 


•9 


26.6.13 


8?50 


-2 


•4 


87 80 


-2 


•6 


26.6.13 


90 20 


+ 


•5 


8750 


-2 


•9 


27.6.13 
Mean . . , 


89 20 


-0-5 


IOOO9O40 


±1 


'9 


100089 70 


±1-5 



.-. Resistance = 10009 ° 40+ 1000897 ° = 100090 10 cm./sec. at 20° -OC. 

Probable error of observations = 5 parts in 1,000,000. 

Resistance in international ohms (N.P.L.) - 0*0100038 3 at 20° -0 0. 

Difference : — Absolute measure - international measure x 10 9 = 51 ' 8 parts in 100,000. 



Table XXII. — Results of Measurements at 20°'0 C. of an Effective Resistance 

of 0*002 ohm. 



Part A (rings away). 


Part B (rings near). 


Date. 


Rin 

cm./sec. 


Rin 

inter- 
national 
ohms. 


Difference. 

Abs. -int. (10 9 ) 

parts in 

100,000. 


Differ- 
ence 
from 
mean. 


Date. 

28.1.13 

28.1.13 

4.2.13 

5.2.13 

6.2.13 


R in 

cm./sec. 

200469 
199457 9 
200465 4 

468 6 
473 r 


Rin 
inter- 
national 
ohms. 

200365 

199354 

200365 

365 

365 


Ditierence. 

Abs.- int. (10 9 ) 

parts in 

100,000. 


Differ- 
ence 
from 

mean. 


6.3.13 
6.3.13 
6.3.13 
7.3.13 
7.3.13 


200479 7 
472 3 

472 4 
478 6 
478i 


200368 2 
368 2 
368 2 
368 2 
368 3 


55*8 ' 

52-1 

52-1 

55*2 

55-0 


+ 1-8 
-1-9 
-1-9 
+ 1-2 
+ 1-0 

±1-6 


52-0 
52-0 
50-2 
51-8 
54-3 

52-0 


+ 0-0 
+ 0-0 
-1-8 
-0-2 

+ 2*3 


Mean . . 


54-0 


Mean . . 


±0-'9 



Probable error of observations = 6 parts in a million. 

Difference -.—Absolute measure - International measure x 10 9 - (54*0 + 52*0)/2 - 53*0 parts in 100,000. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 



103 



Table XXIII. — Results of Measurements at 20°'0 0. of a Resistance of 0*001 ohm 
(a) when the Coils on Cylinders 1 and 2 only were employed ; (6) when the 
Coils on Cylinders 3 and 4 only were employed. 



(a) Coils on cylinders 1 and 2 used. 


(b) Coils on cylinders 3 and 4 used. 


Date. 


Ring 
position. 


Rin 

cm./sec. 


Difference 
from mean. 

Parts 
in 100,000. 


Date. 


Ring 
position. 


R in 

cm./sec. 


Difference 
from mean. 

Parts 
in 100,000. 


15.4.13 
15.4.13 

17.6.13 
18.6.13 


Away 

i) 

Mean . . . 
Near 

Mean . . . 


100089 7 

87 4 


1-1 
1-2 

o-i 

0-2 


15.4.13 
15.4.13 

17.6.13 
18.6.13 

1 


Away 

Mean . . . 

Near 
» 

Mean . . . 


100093 7 

95 9 


1 • 1 
1-1 

0-4 
0-3 


100088 6 


100094 8 


100092i 

92 4 


100089i 
884 


100092 2 


100088 7 


Grand mean = 1000904 cm./sec. 


Grand mean = 100091s cm./sec. 


Resistance in international ohms = 0* 010003 9 3 . 

Difference : — Absolute value - international 

value x 10 9 = 51 • 1 parts in 100,000. 


Resistance in international ohms = 0*0100039 3 . 

Difference : — Absolute value - international 

value x 10 9 = 52*5 parts in 100,000. 



Table XXIV. 



No. of 
observations 

in each 

position of 

coils. 


" 


Standard. 


Resistanc< 

Absolute 
measure. 


In international 
ohms. 


Difference. 

Parts in 100,000 

fobs, x 10~ 9 ) - (int.). 


28 

12 

5 

2 

2 


Coils 1 and 2 used 
Coils 3 and 4 used 


ohm. 

o-ooi 
o-oi 

0-002 

o-ooi 
o-ooi 


cm./sec. 
100091 6 
100090io 

See Tab 
1000904 
100091 8 


0- 00100039 3 
0- 0100038 3 
le XXII. 
0- 00100039 3 
0- 00100039s 


52-s 

51-s 
53- 

51-i 

52-s 



The agreement is most satisfactory. Weighted mean . . 52 



104 ME. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 

error arising from the observation of the galvanometer deflections. The probable 
observational error of the mean value in absolute measure of the 0*001 ohm resistance 
standard is about 3 parts in 1,000,000. 

The probable error of the diametral dimensions of the coils has been estimated in 
Section 13 to be not greater than 1/x plus the errors of the gauges employed. 
Inspection of the results given in Table III. shows that the mean value taken as 
correct differs from the results obtained in February and March, 1912, by about 1'4/x, 
and from those obtained in April, 1913, by about 0°7ju. The gauges employed are 
believed to be accurate within 0*5^, so that the maximum probable error of the 
diametral dimensions is about +l'5/x which corresponds to a probable error in the 
mutual inductance of about 8 parts in 1,000,000. This assumes that the error is of 
the same sign for all the coils. 

The probable error of the mean axial lengths of the coils has been estimated in 
Section 13 to be about 3/x, and a change of this amount in all the coils and in the 
same direction produces a change in the mutual inductance of 2 parts in 1,000,000. 

Any error due to lack of knowledge of the position of the mean diametral planes of 
the coils is practically eliminated by the reversals. An analysis of the plug 
measurements shows that the error cannot be so great as 2 parts in 1,000,000, and 
we may therefore dismiss it from consideration. 

The growth in the dimensions of the coils due to the passage of the current is not 
really large. The mutual inductance of the coils and brush contact circles is nearly 
3 parts in 100,000 greater after the current has been left on, for 20 minutes, but the 
mean mutual inductance is only about 1*5 parts in 100,000 greater than that at the 
start. The error of estimation of the increase must have an exceedingly small effect 
on our final results and need not be considered. 

The mean diametral distance between opposite segments on the discs diminished by 
0*08 mm. during our experiments. Such a change would be a serious one in the old 
form of Lorenz apparatus, but in our instrument the change in the mutual inductance 
due to such a reduction in the mean diameter is less than 4 parts in 1,000,000. 

Evidence in favour of a small probable error is afforded by the satisfactory 
agreement between the results obtained when coils Nos, 1 and 2, and 3 and 4, were 
used independently. The resistance of the O'OOl ohm standard was found to be 
100090 4 cm./sec. when coils Nos. 1 and 2 only were used, and 100091 8 cm./sec. when 
coils Nos. 3 and 4 were used. The difference is 14 parts in 1,000,000 and part of 
this is possibly due to the errors of observations, for only four observations were made 
in each case. 

The electrical method of setting the coils in position has been shown to be subject to 
errors less than 1 part in 1,000,000 ; the magnetic susceptibility of the parts of the 
apparatus, excluding the motor, is too small to be measured with the apparatus at 
our disposal ■ and the effect of the motor on the mutual inductance -of the coils and 
discs has been proved to be negligibly small. The errors of speed cannot be all of one 



RESISTANCE BY A METHOD BASED ON THAT OF LOKENZ. 105 

sign and must be quite negligible apart from a constant error in the clock rate. But 
this latter is clearly impossible in the case of a standard clock the error of which is 
taken daily. The daily rate was small and time comparisons were made with another 
standard clock the daily rate and error of which were also known. At any hour of 
the day the difference between the clocks agreed, within the possible error of the 
observations, with that calculated from the errors and rates of the clocks. This 
agreement is evidence that both clocks were going uniformly or that there was a 
similar want of uniformity in the going of both. Such similarity is very improbable. 
As an additional precaution the resistance observations were made at times ranging 
from 9 a.m. to 6 p.m. but no systematic differences were observed. 

The possibility of error due to our coils being of wire of finite section has not been 
overlooked. The formula developed by J. Viriamu Jones gives the mutual inductance 
when the coils can be treated as infinitely fine helical filaments, and small corrections 
may be necessary. The case of the coils of the Ayrton-Jones current balance was 
examined by Dr. G. F. 0. Searle, F.R.S., # who showed the correction to be negligible 
in that instrument. In the present instance, no special treatment is necessary as the 
mutual inductance calculations made during the work are sufficient to show that no 
correction need be made. To illustrate this, consider the mutual inductance M of a 
helical filament coinciding with the axis of the wire and the nearer brush contact 
circle. Next consider two helical filaments of the same diametral dimensions as the 
previous one, but let one helix be nearer to, and the other farther from, the circle by 
0*25 mm. Table X. shows that the mean mutual inductance of these two helical 
filaments and the circle is greater than that of the central helical filament by 2 parts 
in 1,000,000. The wire with which the coils are wound is about 0*557 mm. in diameter, 
so that strictly we ought to consider filaments 0'28 mm. away from the central one,, 
but in view of the result obtained we think it unnecessary to calculate the small and 
certainly negligible difference. Next consider two circles coaxial with the disc and at 
equal distances from it, and let the diameter of one circle be greater than that of the 
other by 0'556 mm., the mean diameter being 35*88 cm. At distances of 7*3135 cm.. 
and 23'3150 cm. the mean mutual inductance of these two circles and the circumference 
of the disc does not differ by more than 2 parts in 1,000,000 from the mutual inductance 
of the disc and circle of radius equal to the mean of the two previous ones (see 
Table XVI.). The coil of wire may therefore be treated as a helical filament. 

The determination of a resistance in absolute measure is therefore subject to a number 
of small errors, the greatest of which is associated with the determination of the mean 
radius of the coils. This error is probably not greater than 1 part in 100,000, and if 
the remaining errors were all of the same sign it is unlikely that their sum would 
exceed another part in 100,000. 

We believe, therefore, that the absolute measurements of resistance which we have 
made are correct within 2 parts in 100,000. 

* < Phil. Trans.,' A, 207, pp. 541-544. 
VOL. CCXIV. A. P 



106 ME. F. E. SMITH ON THE ABSOLUTE MEASUEEMENTS OF A 

The resistances which we have employed have been compared with nine new mercury 
standards of resistance constructed in accordance with the Specifications of the London 
Conference on Electrical Units and Standards (1908). These mercury standards of 
resistance practically realise the international ohm then defined, and the agreement 
between the nine standards is very good. # There are, however, certain sources of 
error in the construction of such standards which must always be of the same sign, and 
the probable error associated with the practical realisation of the international ohm 
has been estimated to be not less than 2 or 3 parts in 100,000. 

The mean of the results given in Tables XX., XXI., XXII., and XXIII. leads to 
the conclusion : — ■ 

A resistance of 1 international ohm is equal to 1*00052 ±0*00004 ohm (10 9 cm./sec.), 
the probable error ±0*00004 being approximately the sum of those involved in the 
realisation of the ohm and the international ohm. 

The international ohm, as defined by the London Conference on Electrical Units and 
Standards (1908), is the resistance at 0° C. of a column of mercury, 14*4521 gr. in mass 
of a constant cross-sectional area and of a length of 106*300 cm. As stated in the 
Introduction, the cross-section of such a column is equal to 1 sq. mm. or nearly so. 

Since the international ohm is equal to 1*00052 ±0*00004 ohms, the mass of the 
column of mercury of the same cross-sectional area as the international ohm and having 

a resistance of 1 ohm will be ^ — = 14'4446±0'0006gr.,fwhilethe length 

of the column will be 106 ' 300 = 106*245 + 0*00004 cm.f 

r00052±0*00004 

We may sum up our results by stating that :— 

The ohm 10 9 cm./sec. is represented by the resistance at 0° C. of a column of mercury 
14*4446 ±0*0006 gr. in mass, of a constant cross-sectional area (the same as for the 
international ohm) and having a length of 106*245 ±0*004 cm. 

The Historical Introduction shows a number of determinations, notably those of 
Rayleigh (corrected values marked (S) Table I.), Glazebrook (corrected value (S) 
Table I.), Wiedemann, Dorn, and Himstedt, in close agreement with that now 
obtained. These results are as follows : — 

1882, Kayleigh 106*26 cm. 

1882, Glazebrook 106*25 

1883, Kayleigh 106*24 

1885, Wiedemann 106*265 

1889, Dorn 106*243 

1892, Himstedt 106*259 



35 



* ' Keport of the National Physical Laboratory for 1912.' 

t The probable errors in these two values are so related that an error in either value is necessarily 
associated with an equivalent proportionate error in the other. 



RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 107 

On the whole, the remaining results in Table L give the length of the column of 
mercury representing the ohm as greater than 106*25 cm., and while a discussion of 
the methods and apparatus might do much to explain some of the differences, much 
greater detail of the apparatus than is readily accessible is necessary for this to be 
done. 

The ohm and the international ohm differ by about 5 parts in 10,000, and since the 
ampere (lO -1 C.G.S. unit) has been realised with considerable accuracy, we may 
express the electromotive force of the Weston normal cell in absolute measure. The 
value found in 1908 by Ayrton, Mather and Smith at the National Physical 
Laboratory for the E.M.F. of the Weston normal cell in terms of the ampere 
(10" 1 C.G.S. unit) and the international ohm is 1*01830 volts at 17° C, with a 
probable error of 2 parts in 100,000. The deduced value at 20° C is 1*01818 volts. 
Since that time measurements have been made by Janet, Laporte, and Jouast # at 
the Laboratoire Central d'Electricite ; by Prof. GuiLLETf and Prof. Pell at, f by 
Haga,J and by Rosa and Dorsey§ at the Bureau of Standards. The results are : — ■ 

Ayrton, Mather, and Smith 1*01818 

Janet, Laporte, and Jotj ast 1*01836 

Gijillet . . , , . . . 1*01812 

Pellat . . . 1*01831 

Haga 1*01825 

Rosa and Dorsey 1*01822 

Mean 1*01824 at 20° C. 



These results include the errors of the resistance standards employed as well as the 
errors of the determination of the current in absolute measure. In addition small 
differences existed between the Weston cells. Considering all these circumstances 
the agreement is very remarkable and testifies to the great care taken in the measure- 
ments. There appears to be little doubt that the value 1*01824 at 20° C. is correct, 
within 1 part in 10,000. We conclude that the value of the cell in volts (10 8 C.G.S. 
units) is 1*0188 ± '0001 at 20° C. This value may serve for the present for those 
engaged in absolute measurements. 

Section 24. — Conclusions. 

From the measurements recorded in the previous sections, we conclude that the 
instrument we have described may be used for absolute measurements of resistance 

* 'Bull, de la Soc. Internat. des Electriciens ? (2), vol. 8, p. 459, 1908, and ' Comptes Rendus/ 153, 
p. 718, 1911. 

t 'Bull, de la Soc. Internat. des Electriciens,' 1908. 

| < Konink Akad. Wetensch. Amsterdam Proc.,' p. 587, 1910. 

§ 'Bull. Bureau of Standards/ vol. 8, p. 269, 1912. 

P 2 



108 ME. F. E. SMITH: ABSOLUTE MEASUREMENTS OF A RESISTANCE, ETC. 

with a precision satisfying all present demands whether purely scientific or technical. 
We believe the instrument fully realises the desires of those who were responsible for 
such measurements being made, and the results justify Lord Rayleigh's belief that 
the ohm, as defined in absolute measure, can be realised with a precision comparable 
with that of the international ohm. The instrument can be used at any time ; the 
dimensions of all its parts can be checked when desired, the probable error associated 
with a single measurement is small, and the observations do not unduly tax the 
experimenter. We have formed an estimate, based on the measurements already 
made, of the time necessary to devote to a complete re-determination supposing the 
cylinders to be stripped of the coils. The re-winding of the coils would occupy three 
days, the metrology measurements would extend over six days, and sufficient 
resistance observations could be made in eight more days. Absolute measurements of 
resistance may therefore be placed in the front rank of precision measurements. 

In conclusion, we desire to express our sincere thanks to the Drapers Company of 
London for providing a large sum of money towards the cost of the instrument, and 
to Sir Andrew Noble, F.R.S., for his generous help with the heavy metal work. 

Our most hearty thanks are hereby tendered to our Director, Dr. R. T. Glazebrook, 
C.B., F.R.S., who has not only given his very valuable help and advice throughout 
the work, but has fully appreciated the many difficulties which have arisen and which 
required much time and patience to remove. 

Our best thanks are also due to Lord Rayleigh for his keen interest in the 
investigation, to Dr. Stanton for superintending the turning of the marble 
cylinders, and to many of the staff of the National Physical Laboratory, particularly 
Mr. A. Campbell and Mr. Dye, for suggestive aid throughout the investigation. 




Fig. 3. 




Fig. 4. 



To face p. 40.